Finite Group Databasebeta

← #33137553

Subgroups — #33137553

Order 18688.

862 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988991C211yesmaximal, characteristic, complemented
2C2×#17988865C211yesmaximal, characteristic
3#22614882C211yesmaximal, characteristic
4#22612984C211yesmaximal, characteristic, complemented
5#22611406C211yesmaximal, characteristic, complemented
6#22611402C211yesmaximal, characteristic, complemented
7#22611317C211yesmaximal, characteristic
8#22610881C211yesmaximal, characteristic, complemented
9#22610690C211yesmaximal, characteristic, complemented
10#22610686C211yesmaximal, characteristic, complemented
11#22610646C211yesmaximal, characteristic, complemented
12#22610554C211yesmaximal, characteristic
13#22610548C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#2067C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556590C2211yescharacteristic
18C2×#14556568—21no—
19C2×#14556568—21no—
20C2×#14556556—21no—
21C2×D4×Dic73C2211yescharacteristic
22#17988991—24no—
23#17988865—24no—
24C2×#14556525C2211yescharacteristic
25C2×#14556525C2211yescharacteristic
26C2×#14556524C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556524C2211yescharacteristic
29C2×#14556520C2211yescharacteristic
30C2×#14556520C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34#17988523—21no—
35#17988523—21no—
36#17988491—21no—
37#17988490C2211yescharacteristic
38#17988483C2211yescharacteristic
39#17988469—21no—
40#17988468—21no—
41#17988457C2211yescharacteristic
42#17988456C2211yescharacteristic
43C4⋊C4×Dic73C2211yescharacteristic
44C4⋊C4×Dic73C2211yescharacteristic
45C4×#11924007C2211yescharacteristic
46C4×C4⋊Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53#17988341C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58#390×C73C2211yescharacteristic
59#389×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×Dic146—21no—
63C23×Dic73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556590—22no—
67#14556590—22no—
68#14556590—22no—
69#14556590D414yes—
70#14556568—44no—
71#14556568—44no—
72#14556556—44no—
73D4×Dic73—22no—
74D4×Dic73—22no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007D411yescharacteristic
87C2×#11924007D411yescharacteristic
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2311yescharacteristic
96#14556525—22no—
97#14556525—22no—
98#14556524—22no—
99#14556524—22no—
100#14556524—22no—
101#14556520—22no—
102#14556520—22no—
103#14556512—22no—
104#14556512Q814yescomplemented
105C8×Dic73—22no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498D411yescharacteristic
111#14556498D411yescharacteristic
112C22×D4×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#107×C73—22no—
123C4⋊C8×C73—22no—
124#102×C73—22no—
125#102×C73—22no—
126#95×C73—21no—
127C2×Dic146—24no—
128C2×Dic146—42no—
129C2×Dic146—42no—
130C2×Dic146—42no—
131C2×Dic146—42no—
132C2×Dic146—42no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73—42no—
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831C2×Q812yes—
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⋊Dic73C2×D412yes—
159C4⋊Dic73C2×Q812yes—
160#11924007—24no—
161#11924007—22no—
162#11924007—42no—
163#11924007—42no—
164#11924007—42no—
165#11924007—42no—
166#11924007D814yes—
167C22⋊Dic73—42no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—42no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73C2×D412yes—
176C4×Dic73#4112yes—
177C24×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73C2×Q812yes—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2411yescharacteristic
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73#4112yes—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73C2×D412yes—
203C22⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C42×C73—21no—
206C42×C73—21no—
207C23×C73—22no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73C22×D411yescharacteristic
215D4×C73—22no—
216D4×C73—22no—
217D4×C73—41no—
218D4×C73—41no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
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—41no—
233C2×C4×C73C2×D812yes—
234C2×C4×C73C22×D411yescharacteristic
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C732−1+411yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—22no—
245C2×Dic73—21no—
246C2×Dic73—22no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—84no—
260C2×Dic73#12112yes—
261C2×Dic73#12112yes—
262C2×Dic73C2×D812yes—
263C2×Dic73C2×D812yes—
264C2×Dic73AGL(1,ℤ/8)12yes—
265Dic146—48no—
266Dic146—84no—
267C8×C73—22no—
268C8×C73—41no—
269#10596831—22no—
270#10596831—41no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—42no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73—81no—
278C22×C73C2×#12111yescharacteristic
279C22×C73#54711yescharacteristic
280C22×C73D4×Q811yescharacteristic
281C22×C73C22×D811yescharacteristic
282C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C73#57811yescharacteristic
284C22×C73#57811yescharacteristic
285C4×C73—22no—
286C4×C73—41no—
287C4×C73—42no—
288C4×C73—81no—
289C4×C73—81no—
290C4×C73#53311yescharacteristic
291C4×C73#54111yescharacteristic
292C4×C73#54811yescharacteristic
293C4×C73D4×Q811yescharacteristic
294Dic73—22no—
295Dic73—22no—
296Dic73—41no—
297Dic73—42no—
298Dic73—42no—
299Dic73—81no—
300Dic73—81no—
301Dic73#45914yes—
302#25449—731nomaximal, Sylow
303C2×C73—82no—
304C2×C73#242911yescharacteristic
305C2×C73C2×#45911yescharacteristic
306C2×C73#304411yescharacteristic
307C2×C73#333411yescharacteristic
308C2×C73#335311yescharacteristic
309C2×C73Q8×D811yescharacteristic
310C2×C73#338611yescharacteristic
311C2×#500—731no—
312C2×#474—731no—
313#2542—731no—
314#2365—731no—
315#2083—731no—
316#2083—731no—
317#2067—731no—
318#2067D7311yescharacteristic
319#1988—731no—
320#1947—731no—
321#1947—731no—
322#1939—731no—
323#1925—731no—
324#1925—731no—
325C4×#102—731no—
326C73#2544911yescharacteristic, complemented, Sylow
327C2×C4⋊Q8—731no—
328C2×C4⋊Q8—731no—
329C2×C22⋊Q8—1461no—
330C2×C22⋊Q8—1461no—
331C2×C4×Q8—1461no—
332C2×C4×D4—731no—
333#500—1464no—
334#474—1464no—
335C2×#107—731no—
336C2×#107—731no—
337C2×#107—731no—
338C2×#107—731no—
339C2×#107—731no—
340C2×#107C2×D7311yescharacteristic
341C2×C4⋊C8—731no—
342C2×C4⋊C8—731no—
343C2×C4⋊C8C2×D7311yescharacteristic
344C2×#102—731no—
345C2×#102—731no—
346C2×#102C2×D7311yescharacteristic
347C2×#102C2×D7311yescharacteristic
348C2×C4×C8—731no—
349#399—1461no—
350#399—1461no—
351#391—1461no—
352#390—731no—
353#390C2×D7311yescharacteristic
354#389—731no—
355#389C2×D7311yescharacteristic
356#387—1461no—
357#387—1461no—
358#384—731no—
359#384—731no—
360C4×C4⋊C4—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4—731no—
363C4×C4⋊C4—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340C2×D7311yescharacteristic
370#336—731no—
371#336—731no—
372C22×Q8—1461no—
373C22×Q8—1461no—
374C22×D4C22×D7311yescharacteristic
375C23×C4—1461no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8C22×D7311yescharacteristic
379C22×C8C22×D7311yescharacteristic
380C4⋊Q8—734no—
381C4⋊Q8—1462no—
382C4⋊Q8—1462no—
383C4⋊Q8—1462no—
384C22⋊Q8—2924no—
385C22⋊Q8—2924no—
386C4×Q8—2924no—
387C4×D4—1462no—
388C4×D4—1462no—
389C2×C4⋊C4—21no—
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⋊C4—1461no—
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1461no—
409C2×C22⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—1461no—
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42C22×D7311yescharacteristic
417#107—22no—
418#107—1462no—
419#107—1462no—
420#107—1462no—
421#107—1462no—
422#107—1462no—
423C4⋊C8—22no—
424C4⋊C8—1462no—
425C4⋊C8—1462no—
426#102—22no—
427#102—22no—
428#102—734no—
429#102—1462no—
430C4×C8—1462no—
431#95—21no—
432#95—731no—
433#95—731no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438C24—21no—
439C2×Q8—2922no—
440C2×Q8—2922no—
441C2×Q8—2922no—
442C2×Q8—2922no—
443C2×Q8—2922no—
444C2×Q8—1464no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4Q8×D7312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—2922no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—1461no—
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—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8#1192402712yes—
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—2922no—
491C4⋊C4—2922no—
492C4⋊C4—734no—
493C4⋊C4—732no—
494C4⋊C4—732no—
495C4⋊C4—1462no—
496C4⋊C4—1464no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4D4×D7312yes—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—2922no—
507C22⋊C4—2922no—
508C22⋊C4—2922no—
509C42—21no—
510C42—21no—
511C42—2922no—
512C42—2922no—
513C42—2922no—
514C42—732no—
515C42—732no—
516C42—1461no—
517C42—1461no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D7311yescharacteristic
526Q8—2928no—
527Q8—5844no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—5844no—
557C2×C4—732no—
558C2×C4—732no—
559C2×C4—732no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4D8×D7312yes—
572C2×C4C2×D4×D7311yescharacteristic
573C2×C4C2×Q8×D7311yescharacteristic
574C2×C4#41×D7311yescharacteristic
575C2×C4C2×#1192402711yescharacteristic
576C2×C4#1455663811yescharacteristic
577C2×C4#1455663811yescharacteristic
578C8—22no—
579C8—41no—
580C8—2921no—
581C8—1462no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22#121×D7311yescharacteristic
590C22#1798923711yescharacteristic
591C22#1798925011yescharacteristic
592C22C2×D8×D7311yescharacteristic
593C22AGL(1,ℤ/8)×D7311yescharacteristic
594C22#1798940111yescharacteristic
595C22#1798940311yescharacteristic
596C4—22no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—2921no—
602C4—2922no—
603C4—2922no—
604C4—5841no—
605C4—5841no—
606C4—734no—
607C4—1462no—
608C4—1462no—
609C4#1798913911yescharacteristic
610C4#1798919411yescharacteristic
611C4#1798924311yescharacteristic
612C4#1798925011yescharacteristic
613C2—82no—
614C2#2261352311yescharacteristic
615C2#459×D7311yescharacteristic
616C2#2261945911yescharacteristic
617C2#2262227511yescharacteristic
618C2#2262251011yescharacteristic
619C2#2262278111yescharacteristic
620C2#2262284711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2067C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2067D73
hypocenterC73#25449
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. C2×#17988991C2semidirect
4. #22612984C2semidirect
5. #22611406C2semidirect
6. #22611402C2semidirect
8. #22610881C2semidirect
9. #22610690C2semidirect
10. #22610686C2semidirect
11. #22610646C2semidirect
104. #14556512Q8semidirect
326. C73#25449semidirect

Maximal subgroups

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

subgroupindex
C2×#179889912
C2×#179888652
#226148822
#226129842
#226114062
#226114022
#226113172
#226108812
#226106902
#226106862
#226106462
#226105542
#226105482
#102×Dic732
C73×#20672
#2544973