Finite Group Databasebeta

← #33107852

Subgroups — #33107852

Order 18688.

914 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988865C211yesmaximal, characteristic
2#22612984C211yesmaximal, characteristic, complemented
3D4×#11924007C211yesmaximal, characteristic
4#22610890C211yesmaximal, characteristic, complemented
5#22610691C211yesmaximal, characteristic, complemented
6#22610667C211yesmaximal, characteristic, complemented
7#22610666C211yesmaximal, characteristic, complemented
8#22610524C211yesmaximal, characteristic, complemented
9#22610521C211yesmaximal, characteristic, complemented
10#22610190C211yesmaximal, characteristic
11#22609797C211yesmaximal, characteristic, complemented
12#22609758C211yesmaximal, characteristic
13#22609756C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#1864C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556556—21no—
18C2×D4×Dic73C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20C22×#11924007—21no—
21#17988865C418yescomplemented
22C2×#14556525C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556502C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988491—21no—
32#17988485C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988469—21no—
36#17988468—21no—
37#17988460—21no—
38#17988459—21no—
39#17988456C2211yescharacteristic
40#17988445—21no—
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×#11924007C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×#106×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146—21no—
61C23×Dic73—21no—
62C23×Dic73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556590—22no—
66#14556590C2×C414yescomplemented
67#14556556—44no—
68D4×Dic73—22no—
69D4×Dic73—22no—
70D4×Dic73—41no—
71D4×Dic73C2×C414yes—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—44no—
83C2×#11924007D411yescharacteristic
84C2×#11924007D411yescharacteristic
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92#14556525C2×C414yescomplemented
93#14556524C2×C414yescomplemented
94#14556522—41no—
95#14556520C2×C414yescomplemented
96#14556512C2×C414yes—
97#14556512Q814yescomplemented
98#14556502—41no—
99#14556501—41no—
100C8×Dic73—22no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×D4×C73C2311yescharacteristic
110C23×C4×C73—21no—
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×D4×C73—22no—
114C4×D4×C73—41no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C42×C73C2311yescharacteristic
121#106×C73—41no—
122#102×C73—22no—
123#102×C73C2×C414yes—
124#95×C73—21no—
125C2×Dic146—24no—
126C2×Dic146—42no—
127C2×Dic146—42no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—24no—
132C22×Dic73—21no—
133C22×Dic73—42no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73—42no—
137C22×Dic73—81no—
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×#10596831C22×C412yes—
147C2×#10596831C2×Q812yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73C22×C412yes—
154C4⋊Dic73C22×C412yes—
155C4⋊Dic73C2×Q812yes—
156#11924007—24no—
157#11924007—22no—
158#11924007—22no—
159#11924007—42no—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163#11924007—81no—
164C22⋊Dic73—24no—
165C22⋊Dic73—42no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C4×Dic73—42no—
171C4×Dic73C22×C412yes—
172C4×Dic73#4112yes—
173C24×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73C22×C412yes—
178C2×D4×C73C2×Q812yes—
179C2×D4×C73#4112yes—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73—81no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73C22×C412yes—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73C22×C412yes—
202C22⋊C4×C73—42no—
203C42×C73—21no—
204C42×C73—21no—
205C23×C73—22no—
206C23×C73—22no—
207C23×C73—22no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73C22×D411yescharacteristic
213D4×C73—22no—
214D4×C73—22no—
215D4×C73—22no—
216D4×C73C4×Q814yescomplemented
217C2×C4×C73—22no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—84no—
233C2×C4×C73C23×C411yescharacteristic
234C2×C4×C73C22×Q811yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73H(5,𝔽2)11yescharacteristic
238C2×C4×C732−1+411yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—22no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—48no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—84no—
261C2×Dic73C4×D412yes—
262C2×Dic73C4×D412yes—
263C2×Dic73AGL(1,ℤ/8)12yes—
264C2×Dic73AGL(1,ℤ/8)12yes—
265Dic146—48no—
266C8×C73—22no—
267C8×C73—41no—
268#10596831—22no—
269#10596831—22no—
270C22×C73—24no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73C2×C4×D411yescharacteristic
278C22×C73#54611yescharacteristic
279C22×C73D4×Q811yescharacteristic
280C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C73#57711yescharacteristic
283C22×C73#57811yescharacteristic
284C4×C73—41no—
285C4×C73—42no—
286C4×C73—41no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73C2×C4×Q811yescharacteristic
290C4×C73#52011yescharacteristic
291C4×C73#55211yescharacteristic
292C4×C73#55411yescharacteristic
293Dic73—22no—
294Dic73—41no—
295Dic73—42no—
296Dic73—42no—
297Dic73—42no—
298Dic73—41no—
299Dic73—81no—
300Dic73#44214yes—
301#22968—731nomaximal, Sylow
302C2×C73—44no—
303C2×C73#236011yescharacteristic
304C2×C73C2×#44211yescharacteristic
305C2×C73#294711yescharacteristic
306C2×C73#332411yescharacteristic
307C2×C73#333011yescharacteristic
308C2×C73#338411yescharacteristic
309C2×C73#338611yescharacteristic
310C2×#474—731no—
311#2365—731no—
312D4×C4⋊C4—731no—
313#1990—731no—
314#1948—731no—
315#1943—731no—
316#1943—731no—
317#1921—731no—
318#1921—731no—
319#1864—731no—
320#1864D7311yescharacteristic
321#1770—731no—
322#1762—731no—
323#1762—731no—
324C4×#102—731no—
325C73#2296811yescharacteristic, complemented, Sylow
326C2×C4⋊Q8—731no—
327C2×C4×Q8—1461no—
328C2×C4×D4—731no—
329C2×C4×D4—731no—
330C2×C4×D4C2×D7311yescharacteristic
331C22×C4⋊C4—1461no—
332#474—738no—
333C2×#107—731no—
334C2×#107—731no—
335C2×#106—731no—
336C2×#106C2×D7311yescharacteristic
337C2×C4⋊C8—731no—
338C2×#102—731no—
339C2×#102—731no—
340C2×#102C2×D7311yescharacteristic
341C2×#102C2×D7311yescharacteristic
342C2×#97—731no—
343C2×#97—731no—
344C2×C4×C8—731no—
345#391—1461no—
346#389—731no—
347#389—731no—
348#389—731no—
349#389C2×D7311yescharacteristic
350#387—1461no—
351#387—1461no—
352#385—1461no—
353#385—1461no—
354#384—731no—
355#380—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340C2×D7311yescharacteristic
368#336—731no—
369#336C2×D7311yescharacteristic
370C22×Q8—1461no—
371C22×D4C22×D7311yescharacteristic
372C23×C4—21no—
373C23×C4—1461no—
374C23×C4—1461no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8C22×D7311yescharacteristic
378C22×C8C22×D7311yescharacteristic
379C4⋊Q8—734no—
380C4⋊Q8—1462no—
381C4×Q8—2924no—
382C4×D4—22no—
383C4×D4—41no—
384C4×D4—2921no—
385C4×D4—734no—
386C4×D4—1462no—
387C4×D4—1462no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—2924no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C42—731no—
409C2×C42—731no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42C22×D7311yescharacteristic
414#107—734no—
415#107—734no—
416#106—41no—
417#106—2921no—
418C4⋊C8—734no—
419#102—22no—
420#102—734no—
421#102—734no—
422#102C4×D7314yes—
423#97—2921no—
424#97—2921no—
425C4×C8—1462no—
426#95—21no—
427#95—731no—
428#95—731no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435C24—21no—
436C2×Q8—2922no—
437C2×Q8—2922no—
438C2×Q8—1464no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D7312yes—
443C2×D4Q8×D7312yes—
444C2×D4#1192402712yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—2922no—
452C22×C4—2922no—
453C22×C4—2922no—
454C22×C4—2922no—
455C22×C4—5841no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—1461no—
461C22×C4—1464no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4D4×D7311yescharacteristic
466C22×C4D4×D7311yescharacteristic
467C22×C4C23×D7311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—732no—
474C2×C8—732no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8C2×C4×D7312yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—2922no—
489C4⋊C4—2922no—
490C4⋊C4—5841no—
491C4⋊C4—732no—
492C4⋊C4—732no—
493C4⋊C4—732no—
494C4⋊C4—1462no—
495C4⋊C4—1462no—
496C4⋊C4—1464no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4C2×C4×D7312yes—
503C22⋊C4—42no—
504C22⋊C4—2922no—
505C22⋊C4—1464no—
506C42—21no—
507C42—21no—
508C42—2922no—
509C42—2922no—
510C42—2922no—
511C42—732no—
512C42—732no—
513C42—1461no—
514C42—1461no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D7311yescharacteristic
523Q8—2928no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4#1455655614yescomplemented
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2928no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—5844no—
555C2×C4—732no—
556C2×C4—732no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1462no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4C22×C4×D7311yescharacteristic
571C2×C4C2×Q8×D7311yescharacteristic
572C2×C4#41×D7311yescharacteristic
573C2×C4C2×#1192402711yescharacteristic
574C2×C4#1455663611yescharacteristic
575C2×C4#1455663811yescharacteristic
576C8—22no—
577C8—41no—
578C8—1462no—
579C8—1462no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D7311yescharacteristic
588C22#1798922411yescharacteristic
589C22#1798925011yescharacteristic
590C22AGL(1,ℤ/8)×D7311yescharacteristic
591C22AGL(1,ℤ/8)×D7311yescharacteristic
592C22#1798939311yescharacteristic
593C22#1798940111yescharacteristic
594C4—41no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—2921no—
600C4—2922no—
601C4—2922no—
602C4—2922no—
603C4—2921no—
604C4—5841no—
605C4—734no—
606C4—1462no—
607C4C2×#1455655611yescharacteristic
608C4#1798906611yescharacteristic
609C4#1798927411yescharacteristic
610C4#1798929111yescharacteristic
611C2—44no—
612C2#2261293111yescharacteristic
613C2#442×D7311yescharacteristic
614C2#2261876711yescharacteristic
615C2#2262215511yescharacteristic
616C2#2262222911yescharacteristic
617C2#2262282511yescharacteristic
618C2#2262284711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1864C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1864D73
hypocenterC73#22968
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #22612984C2semidirect
4. #22610890C2semidirect
5. #22610691C2semidirect
6. #22610667C2semidirect
7. #22610666C2semidirect
8. #22610524C2semidirect
9. #22610521C2semidirect
11. #22609797C2semidirect
21. #17988865C4semidirect
66. #14556590C2×C4semidirect
92. #14556525C2×C4semidirect
93. #14556524C2×C4semidirect
95. #14556520C2×C4semidirect
97. #14556512Q8semidirect
216. D4×C73C4×Q8semidirect
325. C73#22968semidirect
527. D4#14556556semidirect

Maximal subgroups

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

subgroupindex
C2×#179888652
#226129842
D4×#119240072
#226108902
#226106912
#226106672
#226106662
#226105242
#226105212
#226101902
#226097972
#226097582
#226097562
#102×Dic732
C73×#18642
#2296873