Finite Group Databasebeta

← #33149364

Subgroups — #33149364

Order 18688.

858 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988880C211yesmaximal, characteristic, complemented
2#22613775C211yesmaximal, characteristic
3Q8×#11924007C211yesmaximal, characteristic, complemented
4#22611457C211yesmaximal, characteristic
5#22611405C211yesmaximal, characteristic, complemented
6#22611303C211yesmaximal, characteristic
7#22611257C211yesmaximal, characteristic, complemented
8#22610674C211yesmaximal, characteristic, complemented
9#22610495C211yesmaximal, characteristic, complemented
10#22610288C211yesmaximal, characteristic
11#22610288C211yesmaximal, characteristic
12#22610201C211yesmaximal, characteristic, complemented
13#22610185C211yesmaximal, characteristic, complemented
14#22610178C211yesmaximal, characteristic
15C2×#440×C73C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556570—21no—
18C2×Q8×Dic73C2211yescharacteristic
19C2×Q8×Dic73C2211yescharacteristic
20#17988880—24no—
21C2×#14556525C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30#17988510—21no—
31#17988499—21no—
32#17988490C2211yescharacteristic
33#17988490C2211yescharacteristic
34#17988486C2211yescharacteristic
35#17988485—21no—
36#17988485C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988483—21no—
39#17988483C2211yescharacteristic
40#17988460—21no—
41#17988456—21no—
42#17988456C2211yescharacteristic
43C4×#11924007—21no—
44C4×#11924007C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×SD16×C73C2211yescharacteristic
54C2×C4×Q8×C73C2211yescharacteristic
55C2×C4×D4×C73C2211yescharacteristic
56#440×C73—24no—
57C2×#107×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59C2×#102×C73C2211yescharacteristic
60C2×#97×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556590—22no—
65#14556590—41no—
66#14556570—44no—
67Q8×Dic73—22no—
68Q8×Dic73—22no—
69Q8×Dic73—22no—
70Q8×Dic73—41no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007D411yescharacteristic
85C2×#11924007D411yescharacteristic
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556525—22no—
95#14556524—22no—
96#14556524—41no—
97#14556523—22no—
98#14556520—22no—
99#14556515—22no—
100#14556515—22no—
101#14556512—22no—
102#14556512—41no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×Q8×C73C2311yescharacteristic
110C22×D4×C73C2311yescharacteristic
111C23×C4×C73—21no—
112C2×SD16×C73—22no—
113C2×SD16×C73—22no—
114C2×SD16×C73Q814yescomplemented
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C4×Q8×C73—22no—
118C4×Q8×C73—41no—
119C4×D4×C73—22no—
120C4×D4×C73—41no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C22⋊C4×C73—21no—
125C2×C42×C73—21no—
126C2×C42×C73C2311yescharacteristic
127#107×C73—22no—
128#103×C73D414yes—
129#102×C73—22no—
130#97×C73—22no—
131C2×Dic146—42no—
132C2×Dic146—42no—
133C2×Dic146—81no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C4⋊Dic73—24no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73—42no—
158C4⋊Dic73#4112yes—
159#11924007—22no—
160#11924007—22no—
161#11924007—42no—
162#11924007—42no—
163#11924007—42no—
164#11924007—81no—
165#11924007—81no—
166#11924007—81no—
167C22⋊Dic73—42no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C22⋊Dic73—81no—
171C4×Dic73—24no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—42no—
177C24×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73C2×D412yes—
182C2×Q8×C73C2×Q812yes—
183C2×Q8×C73#4112yes—
184C2×D4×C73—21no—
185C2×D4×C73—21no—
186C2×D4×C73—21no—
187C2×D4×C73—21no—
188C2×D4×C73—21no—
189C2×D4×C73C2×Q812yes—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—42no—
194C22×C4×C73—42no—
195C22×C4×C73—81no—
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2411yescharacteristic
199SD16×C73—28no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73C2×D412yes—
205C2×C8×C73C2×Q812yes—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—42no—
210C4⋊C4×C73C2×D412yes—
211C22⋊C4×C73—42no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—42no—
215C23×C73—22no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73C22×D411yescharacteristic
223Q8×C73—22no—
224Q8×C73—22no—
225Q8×C73—22no—
226Q8×C73C22⋊Q814yescomplemented
227D4×C73—22no—
228D4×C73—22no—
229D4×C73—41no—
230D4×C73—41no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—84no—
247C2×C4×C73C22×D411yescharacteristic
248C2×C4×C73C22×Q811yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73H(5,𝔽2)11yescharacteristic
252C2×C4×C732−1+411yescharacteristic
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—22no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73AGL(1,ℤ/8)12yes—
276C2×Dic73#13712yes—
277Dic146—84no—
278C8×C73—22no—
279C8×C73—22no—
280#10596831—41no—
281#10596831—41no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73#52511yescharacteristic
290C22×C73D4211yescharacteristic
291C22×C73D4×Q811yescharacteristic
292C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C73C2×#13711yescharacteristic
294C22×C73#57611yescharacteristic
295C22×C73#57811yescharacteristic
296C4×C73—24no—
297C4×C73—41no—
298C4×C73—42no—
299C4×C73—41no—
300C4×C73—81no—
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#53911yescharacteristic
303C4×C73#54711yescharacteristic
304C4×C73#55411yescharacteristic
305Dic73—22no—
306Dic73—22no—
307Dic73—41no—
308Dic73—42no—
309Dic73—41no—
310Dic73—81no—
311Dic73—81no—
312Dic73—81no—
313#26397—731nomaximal, Sylow
314C2×C73—82no—
315C2×C73#244911yescharacteristic
316C2×C73#311211yescharacteristic
317C2×C73#312411yescharacteristic
318C2×C73#327011yescharacteristic
319C2×C73#327411yescharacteristic
320C2×C73#338511yescharacteristic
321C2×C73#338611yescharacteristic
322C2×#477—731no—
323C2×#440D7311yescharacteristic
324#2452—731no—
325Q8×C4⋊C4—731no—
326#2093—731no—
327#2083—731no—
328#2065—731no—
329#2057—731no—
330#1944—731no—
331#1917—731no—
332#1880—731no—
333#1880—731no—
334#1865—731no—
335#1863—731no—
336#1862—731no—
337C73#2639711yescharacteristic, complemented, Sylow
338C22×SD16C2×D7311yescharacteristic
339C2×C4⋊Q8—731no—
340C2×C22⋊Q8—1461no—
341C2×C4×Q8—731no—
342C2×C4×Q8—731no—
343C2×C4×Q8C2×D7311yescharacteristic
344C2×C4×D4C2×D7311yescharacteristic
345#477—1464no—
346#440—24no—
347C2×#107—731no—
348C2×#107—731no—
349C2×#107—731no—
350C2×#107C2×D7311yescharacteristic
351C2×#106—731no—
352C2×C4⋊C8—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103C2×D7311yescharacteristic
356C2×#102—731no—
357C2×#102—731no—
358C2×#102C2×D7311yescharacteristic
359C2×#97C2×D7311yescharacteristic
360#396—1461no—
361#393—1461no—
362#390—731no—
363#390—731no—
364#389—731no—
365#389—731no—
366#389—731no—
367#389—731no—
368#389—1461no—
369#389—1461no—
370#385—1461no—
371#384—731no—
372#384—1461no—
373C4×C4⋊C4—731no—
374C4×C4⋊C4—1461no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383C22×Q8—1461no—
384C22×Q8C22×D7311yescharacteristic
385C22×D4C22×D7311yescharacteristic
386C23×C4—21no—
387C2×SD16—22no—
388C2×SD16—22no—
389C2×SD16Dic14614yescomplemented
390C22×C8—731no—
391C22×C8—731no—
392C22×C8C22×D7311yescharacteristic
393C22×C8C22×D7311yescharacteristic
394C4⋊Q8—2921no—
395C4⋊Q8—1462no—
396C22⋊Q8—2924no—
397C4×Q8—22no—
398C4×Q8—41no—
399C4×Q8—2921no—
400C4×Q8—1462no—
401C4×Q8—1462no—
402C4×Q8—1462no—
403C4×D4—22no—
404C4×D4—41no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C4⋊C4C22×D7311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4—1461no—
426C2×C22⋊C4—1461no—
427C2×C22⋊C4—1461no—
428C2×C42—21no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42C22×D7311yescharacteristic
434#107—22no—
435#107—2921no—
436#107—1462no—
437#107—1462no—
438#106—1462no—
439C4⋊C8—1462no—
440#103—1462no—
441#103—1462no—
442#103C22⋊D7314yes—
443#102—22no—
444#102—2921no—
445#102—1462no—
446#97—22no—
447#95—731no—
448#95—731no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453C24—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—2922no—
458C2×Q8—2922no—
459C2×Q8—5841no—
460C2×Q8C2×C22⋊D7312yes—
461C2×Q8C2×Dic14612yes—
462C2×Q8#1192402812yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic14612yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4D4×D7311yescharacteristic
484C22×C4D4×D7311yescharacteristic
485C22×C4C23×D7311yescharacteristic
486SD16—28no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8C2×C22⋊D7312yes—
498C2×C8C2×Dic14612yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—2922no—
504C4⋊C4—2922no—
505C4⋊C4—2922no—
506C4⋊C4—2922no—
507C4⋊C4—2922no—
508C4⋊C4—5841no—
509C4⋊C4—5841no—
510C4⋊C4—5841no—
511C4⋊C4—732no—
512C4⋊C4—1462no—
513C4⋊C4—1462no—
514C4⋊C4—1464no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4C2×C22⋊D7312yes—
523C22⋊C4—42no—
524C22⋊C4—2922no—
525C22⋊C4—2922no—
526C22⋊C4—2922no—
527C22⋊C4—5841no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—2922no—
532C42—1464no—
533C42—1461no—
534C42—1461no—
535C42—1461no—
536C42—1461no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D7311yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—5844no—
549Q8#1455657014yescomplemented
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—732no—
582C2×C4—732no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1462no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4C22×C22⋊D7311yescharacteristic
595C2×C4C22×Dic14611yescharacteristic
596C2×C4#41×D7311yescharacteristic
597C2×C4C2×#1192402811yescharacteristic
598C2×C4#1455663611yescharacteristic
599C2×C4#1455663811yescharacteristic
600C8—22no—
601C8—22no—
602C8—2921no—
603C8—2921no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1798909611yescharacteristic
612C22D4×C22⋊D7311yescharacteristic
613C22D4×Dic14611yescharacteristic
614C22AGL(1,ℤ/8)×D7311yescharacteristic
615C22#137×D7311yescharacteristic
616C22#1798939111yescharacteristic
617C22#1798940311yescharacteristic
618C4—24no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—81no—
623C4—2922no—
624C4—2921no—
625C4—2921no—
626C4—5841no—
627C4—5841no—
628C4—5841no—
629C4—1462no—
630C4—1462no—
631C4C2×#1455657011yescharacteristic
632C4#1798917911yescharacteristic
633C4#1798923611yescharacteristic
634C4#1798929611yescharacteristic
635C2—82no—
636C2#2261373411yescharacteristic
637C2#2262005111yescharacteristic
638C2#2262019111yescharacteristic
639C2#2262149511yescharacteristic
640C2#2262154511yescharacteristic
641C2#2262284211yescharacteristic
642C2#2262285311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#440×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenterC2×#440D73
hypocenterC73#26397
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988880C2semidirect
3. Q8×#11924007C2semidirect
5. #22611405C2semidirect
7. #22611257C2semidirect
8. #22610674C2semidirect
9. #22610495C2semidirect
12. #22610201C2semidirect
13. #22610185C2semidirect
114. C2×SD16×C73Q8semidirect
226. Q8×C73C22⋊Q8semidirect
337. C73#26397semidirect
389. C2×SD16Dic146semidirect
549. Q8#14556570semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888802
#226137752
Q8×#119240072
#226114572
#226114052
#226113032
#226112572
#226106742
#226104952
#226102882
#226102012
#226101852
#226101782
C2×#440×C732
#2639773