Finite Group Databasebeta

← #33242955

Subgroups — #33242955

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622279C211yesmaximal, characteristic, complemented
2#22612167C211yesmaximal, characteristic, complemented
3#22612126C211yesmaximal, characteristic, complemented
4#22612114C211yesmaximal, characteristic, complemented
5#22612078C211yesmaximal, characteristic, complemented
6#22612032C211yesmaximal, characteristic, complemented
7#22612029C211yesmaximal, characteristic, complemented
8#22611983C211yesmaximal, characteristic, complemented
9#22611854C211yesmaximal, characteristic, complemented
10#22611843C211yesmaximal, characteristic, complemented
11#22611805C211yesmaximal, characteristic, complemented
12#22611757C211yesmaximal, characteristic, complemented
13#22611643C211yesmaximal, characteristic, complemented
14C2×#17988388C211yesmaximal, characteristic, complemented
15C73×#3280C211yesmaximal, characteristic, complemented
16#17989241—21no—
17C2×#14556534C2211yescharacteristic, complemented
18C2×#14556532C2211yescharacteristic
19C2×#14556532C2211yescharacteristic
20C2×#14556531C2211yescharacteristic, complemented
21C22×#11924003C2211yescharacteristic
22#17988992C2211yescharacteristic, complemented
23#17988901—21no—
24#17988901—21no—
25#17988868—21no—
26#17988823C2211yescharacteristic
27#17988823C2211yescharacteristic, complemented
28C4×D584C2211yescharacteristic, complemented
29C2×#14556525C2211yescharacteristic
30C2×#14556525C2211yescharacteristic, complemented
31#17988414C2211yescharacteristic, complemented
32#17988414C2211yescharacteristic, complemented
33#17988414C2211yescharacteristic, complemented
34C4⋊#11924003C2211yescharacteristic, complemented
35#17988391C2211yescharacteristic, complemented
36#17988391C2211yescharacteristic, complemented
37#17988391C2211yescharacteristic, complemented
38#17988388C2211yescharacteristic
39#17988388C2211yescharacteristic
40#17988388C2211yescharacteristic
41#17988388C2211yescharacteristic
42#17988388C2211yescharacteristic
43#17988388C2211yescharacteristic
44#17988387C2211yescharacteristic, complemented
45C22⋊#11924003C2211yescharacteristic
46C22⋊#11924003C2211yescharacteristic, complemented
47#17988363C2211yescharacteristic, complemented
48C2×#135×C73C2211yescharacteristic
49C22×D8×C73C2211yescharacteristic
50#546×C73—21no—
51#495×C73C2211yescharacteristic
52#467×C73C2211yescharacteristic
53#466×C73C2211yescharacteristic
54#463×C73—21no—
55#450×C73—21no—
56#449×C73—21no—
57C4×D8×C73C2211yescharacteristic
58C8×D4×C73C2211yescharacteristic
59C2×#11924027—41no—
60C2×D584C2311yescharacteristic
61#14556591—21no—
62#14556575—41no—
63#14556575—41no—
64#14556571—41no—
65#14556565—21no—
66#14556565—21no—
67C4×D292—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70#14556534—41no—
71#14556532—22no—
72#14556532—41no—
73#14556531—41no—
74C2×#11924003—41no—
75C2×#11924003C2311yescharacteristic
76C2×#11924003C2311yescharacteristic
77C2×#11924003C2311yescharacteristic
78C2×#11924003C2311yescharacteristic
79#14556525—21no—
80#14556525—21no—
81#14556525—21no—
82#14556525C2311yescharacteristic
83#14556525C2311yescharacteristic
84#14556525C2311yescharacteristic
85#14556513—21no—
86#14556513—21no—
87#14556513—21no—
88C2×#41×C73—21no—
89C22×D4×C73—21no—
90#135×C73—22no—
91#135×C73—41no—
92C2×Q16×C73C2311yescharacteristic
93C2×SD16×C73—21no—
94C2×SD16×C73—21no—
95C2×D8×C73—22no—
96C2×D8×C73—21no—
97C2×D8×C73—41no—
98C2×D8×C73C2311yescharacteristic
99C2×D8×C73C2311yescharacteristic
100C22×C8×C73C2311yescharacteristic
101C22×C8×C73C2311yescharacteristic
102#124×C73—21no—
103#123×C73—41no—
104C22⋊Q8×C73—21no—
105#121×C73—21no—
106#121×C73—41no—
107#120×C73—41no—
108C4×D4×C73—21no—
109C4×D4×C73D411yescharacteristic
110C2×C22⋊C4×C73—41no—
111#107×C73C2311yescharacteristic
112C4⋊C8×C73D411yescharacteristic
113#103×C73—21no—
114#102×C73—21no—
115#102×C73—21no—
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119#11924027—84no—
120C2×Dic146—41no—
121C2×D292—21no—
122C2×C22⋊D73—41no—
123C2×C22⋊D73—41no—
124C22×Dic73—41no—
125C22×Dic73—41no—
126C2×C4×D73—41no—
127D584—21no—
128D584—21no—
129C4⋊Dic73—21no—
130C4⋊Dic73—21no—
131C4⋊Dic73—21no—
132C4⋊Dic73—41no—
133C4⋊Dic73—41no—
134C4⋊Dic73—41no—
135#11924005—41no—
136#11924005—41no—
137#11924005—41no—
138#11924003—21no—
139#11924003—21no—
140#11924003—21no—
141#11924003—21no—
142C24×C73—41no—
143#41×C73—42no—
144#41×C73—81no—
145C2×Q8×C73—21no—
146C2×D4×C73—21no—
147C2×D4×C73—21no—
148C2×D4×C73—42no—
149C2×D4×C73—41no—
150C2×D4×C73—41no—
151C2×D4×C73—41no—
152C2×D4×C73—81no—
153C2×D4×C73D811yescharacteristic
154C22×C4×C73—41no—
155C22×C4×C73—41no—
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158Q16×C73—21no—
159Q16×C73—21no—
160SD16×C73—42no—
161SD16×C73—81no—
162D8×C73—21no—
163D8×C73—21no—
164D8×C73—21no—
165D8×C73—42no—
166D8×C73#4112yescomplemented
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73—21no—
170C2×C8×C73—21no—
171C2×C8×C73C2×D411yescharacteristic
172C2×C8×C73C2×D411yescharacteristic
173C2×C8×C73C2×D411yescharacteristic
174C2×C8×C73C2411yescharacteristic
175C4⋊C4×C73—21no—
176C4⋊C4×C73—41no—
177C4⋊C4×C73D811yescharacteristic
178C22⋊C4×C73—41no—
179C22⋊C4×C73—41no—
180C22⋊C4×C73—41no—
181C22⋊C4×C73—41no—
182C22⋊C4×C73—41no—
183C22⋊C4×C73D811yescharacteristic
184C22⋊C4×C73D811yescharacteristic
185C42×C73C2×D411yescharacteristic
186C23×C73—41no—
187C23×C73—41no—
188C23×C73—82no—
189C23×C73—82no—
190C23×C73—161no—
191C23×C73C2×D811yescharacteristic
192C23×C73C2×D811yescharacteristic
193C22×D73—41no—
194Q8×C73—41no—
195Q8×C73—41no—
196D4×C73—22no—
197D4×C73—22no—
198D4×C73—41no—
199D4×C73—41no—
200D4×C73—41no—
201D4×C73—82no—
202D4×C73—82no—
203D4×C73—161no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—82no—
211C2×C4×C73—161no—
212C2×C4×C73C2×D811yescharacteristic
213C2×C4×C73C2×D811yescharacteristic
214C2×C4×C73C2×D811yescharacteristic
215C2×C4×C73C2×D811yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C22⋊D73—82no—
218C22⋊D73—82no—
219C2×Dic73—41no—
220C2×Dic73—41no—
221C2×Dic73—41no—
222C2×Dic73—82no—
223C2×Dic73—82no—
224D292—41no—
225D292—41no—
226C4×D73—82no—
227Dic146—82no—
228C8×C73—21no—
229C8×C73—21no—
230C8×C73—21no—
231C8×C73C2×#4111yescharacteristic
232C8×C73H(5,𝔽2)11yescharacteristic
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73—42no—
239C22×C73—81no—
240C22×C73—81no—
241C22×C73—81no—
242C22×C73—81no—
243C22×C73—81no—
244C22×C73—164no—
245C22×C73#50912yes—
246C22×C73C22×D811yescharacteristic
247C2×D73—81no—
248C2×D73—81no—
249C2×D73—81no—
250C4×C73—21no—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—21no—
254C4×C73—81no—
255C4×C73—81no—
256C4×C73#54611yescharacteristic
257C4×C73#57711yescharacteristic
258Dic73—81no—
259Dic73—81no—
260Dic73—81no—
261#33873—731nomaximal, Sylow
262C2×C73—22no—
263C2×C73—41no—
264C2×C73—82no—
265C2×C73—161no—
266C2×C73#329411yescharacteristic
267C2×C73C2×#50911yescharacteristic
268C2×C73#341611yescharacteristic
269D73—161no—
270#3334—731no—
271#3280D7311yescharacteristic, complemented
272#2243—731no—
273#2235—731no—
274#2233—731no—
275#2226—731no—
276#2218—731no—
277#2217—731no—
278#2210—731no—
279#2186—731no—
280#2184—731no—
281#2177—731no—
282#2168—731no—
283#2139—731no—
284C2×#357—731no—
285C73#3387311yescharacteristic, complemented, Sylow
286C2×#135C2×D7311yescharacteristic
287C22×D8C2×D7311yescharacteristic
288#547—1461no—
289#546—21no—
290C2×SD32—731no—
291C2×D16—731no—
292C2×D16—731no—
293C2×M32—731no—
294C22×C16—731no—
295#500—731no—
296#495C2×D7311yescharacteristic
297#480—1461no—
298#480—1461no—
299#474—1461no—
300#467C2×D7311yescharacteristic
301#466—731no—
302#466—731no—
303#466C2×D7311yescharacteristic
304#463—21no—
305#450—21no—
306#449—21no—
307C4×D8—731no—
308C4×D8C2×D7311yescharacteristic
309C8×D4C2×D7311yescharacteristic
310C2×#107—731no—
311C2×#107—731no—
312#367—731no—
313#367—731no—
314#367—731no—
315C4⋊C16—731no—
316#358—731no—
317#358—731no—
318#358—731no—
319#357—731no—
320#357—731no—
321#357—731no—
322#357—731no—
323#357—731no—
324#357—731no—
325#357—731no—
326C22⋊C16—731no—
327C22⋊C16—731no—
328#346—731no—
329C2×#41—21no—
330C2×#41—2921no—
331C22×D4—21no—
332#135—22no—
333#135—41no—
334C2×Q16C22×D7311yescharacteristic
335C2×SD16—21no—
336C2×SD16—21no—
337C2×D8—22no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—731no—
341C2×D8C22×D7311yescharacteristic
342C2×D8C22×D7311yescharacteristic
343C22×C8C22×D7311yescharacteristic
344C22×C8C22×D7311yescharacteristic
345C4⋊Q8—1461no—
346#124—21no—
347#123—41no—
348#123—2921no—
349#123—2921no—
350C22⋊Q8—21no—
351C22⋊Q8—2921no—
352#121—21no—
353#121—41no—
354#121—1461no—
355#121—1461no—
356#120—41no—
357C4×D4—21no—
358C4×D4—1461no—
359C4×D4C22⋊D7311yescharacteristic
360C2×C4⋊C4—1461no—
361C2×C4⋊C4—1461no—
362C2×C22⋊C4—41no—
363SD32—2921no—
364D16—2921no—
365D16—1462no—
366M32—2921no—
367C2×C16—2921no—
368C2×C16—731no—
369C2×C16—731no—
370C2×C16—731no—
371C2×C16—731no—
372#107—731no—
373#107—731no—
374#107—731no—
375#107—1461no—
376#107—1461no—
377#107—1461no—
378#107C22×D7311yescharacteristic
379C4⋊C8C22⋊D7311yescharacteristic
380#103—21no—
381#102—21no—
382#102—21no—
383#102—1461no—
384#102—1461no—
385#102—1461no—
386C22⋊C8C22⋊D7311yescharacteristic
387C22⋊C8C22⋊D7311yescharacteristic
388C4×C8C22×D7311yescharacteristic
389C24—41no—
390#41—42no—
391#41—81no—
392#41—5844no—
393C2×Q8—21no—
394C2×Q8—2921no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—81no—
402C2×D4—2921no—
403C2×D4—2921no—
404C2×D4—1461no—
405C2×D4#1192401311yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—2921no—
409C22×C4—2921no—
410C22×C4—2921no—
411C22×C4C2×C22⋊D7311yescharacteristic
412C22×C4C2×C22⋊D7311yescharacteristic
413Q16—21no—
414Q16—21no—
415SD16—42no—
416SD16—81no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—42no—
421D8—1461no—
422D8—1461no—
423D8#1192402712yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D7311yescharacteristic
429C2×C8C2×C22⋊D7311yescharacteristic
430C2×C8C2×C22⋊D7311yescharacteristic
431C2×C8C23×D7311yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—2921no—
435C4⋊C4—2921no—
436C4⋊C4—2921no—
437C4⋊C4—1461no—
438C4⋊C4—1461no—
439C4⋊C4—1461no—
440C4⋊C4#1192401311yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—2921no—
447C22⋊C4—2921no—
448C22⋊C4—2921no—
449C22⋊C4#1192401311yescharacteristic
450C22⋊C4#1192401311yescharacteristic
451C42C2×C22⋊D7311yescharacteristic
452C16—1461no—
453C16—1461no—
454C16—1461no—
455C16—1461no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23—82no—
460C23—161no—
461C23—2921no—
462C23C2×#1192401311yescharacteristic
463C23C2×#1192401311yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—5842no—
467D4—22no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—82no—
473D4—82no—
474D4—161no—
475D4—2921no—
476D4—2921no—
477D4—5842no—
478D4—5842no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—161no—
487C2×C4—2921no—
488C2×C4—2921no—
489C2×C4—2921no—
490C2×C4—5842no—
491C2×C4—5842no—
492C2×C4—5842no—
493C2×C4C2×#1192401311yescharacteristic
494C2×C4C2×#1192401311yescharacteristic
495C2×C4C2×#1192401311yescharacteristic
496C2×C4C2×#1192401311yescharacteristic
497C2×C4C22×C22⋊D7311yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8C2×#1192402711yescharacteristic
502C8#1455663611yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—164no—
515C22—5841no—
516C22—5841no—
517C22—5841no—
518C22#1798902212yes—
519C22C22×#1192401311yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—5841no—
527C4—5841no—
528C4—5841no—
529C4#1798923111yescharacteristic
530C4#1798939811yescharacteristic
531C2—22no—
532C2—41no—
533C2—82no—
534C2—161no—
535C2—11681no—
536C2#2262174411yescharacteristic
537C2C2×#1798902211yescharacteristic
538C2#2262307811yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3280C2
socleC22×C73C22×D8
hypercenter#3280D73
hypocenterC73#33873
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #22622279C2semidirect
2. #22612167C2semidirect
3. #22612126C2semidirect
4. #22612114C2semidirect
5. #22612078C2semidirect
6. #22612032C2semidirect
7. #22612029C2semidirect
8. #22611983C2semidirect
9. #22611854C2semidirect
10. #22611843C2semidirect
11. #22611805C2semidirect
12. #22611757C2semidirect
13. #22611643C2semidirect
14. C2×#17988388C2semidirect
15. C73×#3280C2semidirect
17. C2×#14556534C22semidirect
20. C2×#14556531C22semidirect
22. #17988992C22semidirect
27. #17988823C22semidirect
28. C4×D584C22semidirect
30. C2×#14556525C22semidirect
31. #17988414C22semidirect
32. #17988414C22semidirect
33. #17988414C22semidirect
34. C4⋊#11924003C22semidirect
35. #17988391C22semidirect
36. #17988391C22semidirect
37. #17988391C22semidirect
44. #17988387C22semidirect
46. C22⋊#11924003C22semidirect
47. #17988363C22semidirect
166. D8×C73#41semidirect
271. #3280D73semidirect
285. C73#33873semidirect
423. D8#11924027semidirect

Maximal subgroups

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

subgroupindex
#226222792
#226121672
#226121262
#226121142
#226120782
#226120322
#226120292
#226119832
#226118542
#226118432
#226118052
#226117572
#226116432
C2×#179883882
C73×#32802
#3387373