Finite Group Databasebeta

← #33068886

Subgroups — #33068886

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620307C211yesmaximal, characteristic
2C2×#17988787C211yesmaximal, characteristic
3Q8×#11924012C211yesmaximal, characteristic, complemented
4#22609501C212yesmaximal, complemented
5#22609473C212yesmaximal, complemented
6#22609377C212yesmaximal, complemented
7#22609349C212yesmaximal, complemented
8#22608684C211yesmaximal, characteristic, complemented
9#22608621C211yesmaximal, characteristic, complemented
10#22608598C211yesmaximal, characteristic
11C73×#3104C211yesmaximal, characteristic, complemented
12C22×#11924015—21no—
13C2×#14556588C2211yescharacteristic
14#17988953—22no—
15#17988935C2212yes—
16#17988935C2212yes—
17#17988787—21no—
18#17988787—21no—
19#17988787C2212yes—
20#17988787C2212yes—
21#17988718—22no—
22#17988718C2212yescomplemented
23Q8×#10596831C2212yescomplemented
24#17988611—21no—
25#17988611C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27#17988591—21no—
28#17988591—21no—
29C2×#14556511—21no—
30C4×#11924012—21no—
31C4×#11924012C2211yescharacteristic
32#17988339C2212yescomplemented
33#17988337C2212yescomplemented
34#17988336C2212yescomplemented
35#17988333C2212yescomplemented
36#17988330C2212yes—
37#17988330C2212yes—
38#533×C73C2211yescharacteristic
39C2×C4×Q8×C73C2211yescharacteristic
40#484×C73—21no—
41#477×C73C2211yescharacteristic, complemented
42#477×C73C2211yescharacteristic, complemented
43#462×C73C2211yescharacteristic, complemented
44#462×C73C2211yescharacteristic, complemented
45#451×C73—21no—
46C4×Q16×C73—21no—
47C4×Q16×C73—21no—
48#431×C73C2211yescharacteristic
49C22×D292—21no—
50C22×D292—21no—
51C22×D292—21no—
52C2×#11924015—22no—
53C2×#11924015—22no—
54C2×#11924015—44no—
55C2×#11924012—21no—
56C2×#11924012D411yescharacteristic
57C2×#11924012D411yescharacteristic
58C22×#10596831—21no—
59#14556588—21no—
60#14556588—21no—
61#14556588—21no—
62#14556588C2312yes—
63C4⋊#10596831—22no—
64C4⋊#10596831—22no—
65C4⋊#10596831—21no—
66C4⋊#10596831—21no—
67C4⋊#10596831C2312yes—
68C4⋊#10596831C2312yes—
69#14556511—22no—
70#14556511—22no—
71#14556511—22no—
72#14556511—22no—
73#14556511—22no—
74#14556511—22no—
75#14556502—22no—
76#14556502D412yes—
77C4×#10596831—22no—
78C4×#10596831C2312yes—
79C22×Q8×C73D411yescharacteristic
80C2×Q16×C73—21no—
81C2×Q16×C73—21no—
82C4⋊Q8×C73C2311yescharacteristic
83C4⋊Q8×C73C2311yescharacteristic
84#125×C73—21no—
85C22⋊Q8×C73—21no—
86C22⋊Q8×C73—21no—
87C4×Q8×C73—21no—
88C4×Q8×C73—21no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73D411yescharacteristic
92C4×Q8×C73D411yescharacteristic
93C4×Q8×C73C2311yescharacteristic
94C4×Q8×C73C2311yescharacteristic
95#117×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73D411yescharacteristic
98C2×C42×C73—21no—
99C2×C42×C73C2311yescharacteristic
100#107×C73—21no—
101#107×C73—21no—
102C4⋊C8×C73C2311yescharacteristic
103C4⋊C8×C73C2311yescharacteristic
104#103×C73—21no—
105#103×C73—21no—
106#103×C73—21no—
107#103×C73—21no—
108C22⋊C8×C73—21no—
109C4×C8×C73—21no—
110C23×D73—41no—
111C2×D292—22no—
112C2×D292—22no—
113C2×D292—22no—
114C2×D292—42no—
115C2×D292—42no—
116C2×D292—42no—
117C2×D292—42no—
118C2×D292—41no—
119C2×D292—41no—
120C2×D292—41no—
121C2×D292—82no—
122#11924015—44no—
123#11924015—44no—
124#11924012—22no—
125#11924012—22no—
126#11924012—42no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—42no—
130C2×#10596831C2×D412yes—
131C2×#10596831C2×D412yes—
132C2×Q8×C73—21no—
133C2×Q8×C73—22no—
134C2×Q8×C73—21no—
135C2×Q8×C73—21no—
136C2×Q8×C73—41no—
137C2×Q8×C73C2×D411yescharacteristic
138C2×Q8×C73C2×D411yescharacteristic
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73#4111yescharacteristic
144Q16×C73—42no—
145Q16×C73—42no—
146C2×C8×C73—21no—
147C2×C8×C73—21no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—21no—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—41no—
158C4⋊C4×C73—41no—
159C4⋊C4×C73C2×D411yescharacteristic
160C4⋊C4×C73C2×D411yescharacteristic
161C22⋊C4×C73—41no—
162C42×C73—21no—
163C42×C73—21no—
164C42×C73—21no—
165C42×C73—21no—
166C42×C73—41no—
167C42×C73C2×D411yescharacteristic
168C42×C73C2×D411yescharacteristic
169C42×C73#4111yescharacteristic
170C42×C73C2411yescharacteristic
171C23×C73#12111yescharacteristic
172C22×D73—42no—
173C22×D73—82no—
174C22×D73—82no—
175C22×D73—82no—
176Q8×C73—21no—
177Q8×C73—21no—
178Q8×C73—21no—
179Q8×C73—42no—
180Q8×C73—41no—
181Q8×C73—41no—
182Q8×C73—81no—
183Q8×C73AGL(1,ℤ/8)12yes—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—41no—
191C2×C4×C73—42no—
192C2×C4×C73—41no—
193C2×C4×C73—41no—
194C2×C4×C73—81no—
195C2×C4×C73#12111yescharacteristic
196C2×C4×C73#12111yescharacteristic
197C2×C4×C73#12111yescharacteristic
198C2×C4×C73#13512yes—
199C2×C4×C73#13712yes—
200C2×C4×C73C22×D411yescharacteristic
201C2×C4×C73C22×D411yescharacteristic
202C2×C4×C73C2×#4111yescharacteristic
203D292—42no—
204D292—42no—
205D292—42no—
206D292—42no—
207D292—82no—
208D292—82no—
209D292—82no—
210D292—82no—
211C8×C73—41no—
212C8×C73—41no—
213#10596831—22no—
214#10596831—22no—
215#10596831—42no—
216#10596831—42no—
217C22×C73—21no—
218C22×C73—21no—
219C22×C73#46412yes—
220C22×C73C2×#12111yescharacteristic
221C2×D73—82no—
222C2×D73—82no—
223C2×D73—82no—
224C2×D73—164no—
225C4×C73—21no—
226C4×C73—22no—
227C4×C73—21no—
228C4×C73—21no—
229C4×C73—41no—
230C4×C73—41no—
231C4×C73—41no—
232C4×C73—81no—
233C4×C73—81no—
234C4×C73#46412yes—
235C4×C73C2×#13511yescharacteristic
236C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C73C2×#13711yescharacteristic
238C4×C73#57511yescharacteristic
239#19174—731nomaximal, Sylow
240C2×C73—22no—
241C2×C73C2×#46411yescharacteristic
242C2×C73#305511yescharacteristic
243C2×C73#306811yescharacteristic
244D73—162no—
245#3138—731no—
246#3104D7311yescharacteristic, complemented
247C2×#460—731no—
248Q8×M16—731no—
249#1702—732no—
250#1698—732no—
251#1682—732no—
252#1678—732no—
253#1565—731no—
254#1550—731no—
255#1545—731no—
256C73#1917411yescharacteristic, complemented, Sylow
257C22×SD16—1461no—
258#533C2×D7311yescharacteristic
259C2×#127—731no—
260C2×C4×Q8C2×D7311yescharacteristic
261#490—1462no—
262#486—732no—
263#486—732no—
264#484—21no—
265#477C2×D7311yescharacteristic, complemented
266#477C2×D7311yescharacteristic, complemented
267#462C2×D7311yescharacteristic, complemented
268#462C2×D7311yescharacteristic, complemented
269#460—732no—
270#460—732no—
271#460—1461no—
272#460—1461no—
273#451—21no—
274#446—732no—
275#446—1462no—
276C8×Q8—732no—
277C4×Q16—21no—
278C4×Q16—21no—
279#431C2×D7311yescharacteristic
280#423—731no—
281#423—1461no—
282C2×C4⋊C8—731no—
283#418—1461no—
284#418—1461no—
285C2×#102—1461no—
286C4×M16—731no—
287C4×M16—1461no—
288#335—732no—
289#334—732no—
290#333—732no—
291#332—732no—
292#331—732no—
293#331—732no—
294C22×Q8C22⋊D7311yescharacteristic
295C22×D4—1461no—
296C22×D4—1461no—
297C22×D4—1461no—
298C2×Q16—21no—
299C2×Q16—21no—
300C2×SD16—2924no—
301C2×SD16—1462no—
302C2×SD16—1462no—
303C2×M16—731no—
304C2×M16—731no—
305C2×M16—1461no—
306C22×C8—1461no—
307C4⋊Q8C22×D7311yescharacteristic
308C4⋊Q8C22×D7311yescharacteristic
309#127—732no—
310#127—1461no—
311#127—1461no—
312#127—1461no—
313#125—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D7311yescharacteristic
321C4×Q8C22⋊D7311yescharacteristic
322C4×Q8C22×D7311yescharacteristic
323C4×Q8C22×D7311yescharacteristic
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D7311yescharacteristic
327C2×C42—21no—
328C2×C42C22×D7311yescharacteristic
329#107—21no—
330#107—21no—
331C4⋊C8—732no—
332C4⋊C8—732no—
333C4⋊C8—1462no—
334C4⋊C8—1462no—
335C4⋊C8—1461no—
336C4⋊C8—1461no—
337C4⋊C8C22×D7311yescharacteristic
338C4⋊C8C22×D7311yescharacteristic
339#103—21no—
340#103—21no—
341#103—21no—
342#103—21no—
343#102—1462no—
344#102—1462no—
345#102—1462no—
346#102—1462no—
347#102—1462no—
348#102—1462no—
349C22⋊C8—21no—
350#97—732no—
351#97—1462no—
352C4×C8—21no—
353C4×C8—732no—
354C4×C8—1462no—
355C24—2921no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8C2×C22⋊D7311yescharacteristic
362C2×Q8C2×C22⋊D7311yescharacteristic
363C2×D4—2922no—
364C2×D4—2922no—
365C2×D4—2921no—
366C2×D4—2921no—
367C2×D4—2921no—
368C2×D4—2922no—
369C2×D4—2922no—
370C2×D4—5842no—
371C2×D4—1462no—
372C2×D4—1462no—
373C2×D4—1462no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D7311yescharacteristic
377C22×C4C2×C22⋊D7311yescharacteristic
378C22×C4#1192402711yescharacteristic
379Q16—42no—
380Q16—42no—
381SD16—2924no—
382SD16—2924no—
383M16—2922no—
384M16—1462no—
385M16—1462no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—2922no—
389C2×C8—732no—
390C2×C8—732no—
391C2×C8—1462no—
392C2×C8—1462no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4C2×C22⋊D7311yescharacteristic
405C4⋊C4C2×C22⋊D7311yescharacteristic
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—41no—
412C42D4×D7311yescharacteristic
413C42C2×C22⋊D7311yescharacteristic
414C42#1192402711yescharacteristic
415C42C23×D7311yescharacteristic
416C23—2922no—
417C23—5842no—
418C23—5842no—
419C23—5842no—
420C23#1455656311yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—41no—
426Q8—41no—
427Q8—81no—
428Q8#1455661612yes—
429D4—2922no—
430D4—2922no—
431D4—2922no—
432D4—2922no—
433D4—5842no—
434D4—5842no—
435D4—5842no—
436D4—5842no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4#1455656311yescharacteristic
449C2×C4#1455656311yescharacteristic
450C2×C4#1455656311yescharacteristic
451C2×C4#1455661412yes—
452C2×C4#1455662312yes—
453C2×C4C2×D4×D7311yescharacteristic
454C2×C4C22×C22⋊D7311yescharacteristic
455C2×C4C2×#1192402711yescharacteristic
456C8—41no—
457C8—41no—
458C8—2922no—
459C8—2922no—
460C8—1462no—
461C8—1462no—
462C22—21no—
463C22—21no—
464C22—5842no—
465C22—5842no—
466C22—5842no—
467C22—11684no—
468C22#1798881112yes—
469C22C2×#1455656311yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4#1798881112yes—
480C4C2×#1455661411yescharacteristic
481C4C2×#1455661611yescharacteristic
482C4C2×#1455662311yescharacteristic
483C4#1798938511yescharacteristic
484C2—22no—
485C2—11682no—
486C2C2×#1798881111yescharacteristic
487C2#2261955211yescharacteristic
488C2#2261966111yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3104C2
socleC22×C73C2×#121
prime radicalC2×#14556588C22
hypercenter#3104D73
hypocenterC73#19174
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. Q8×#11924012C2semidirect
4. #22609501C2semidirect
5. #22609473C2semidirect
6. #22609377C2semidirect
7. #22609349C2semidirect
8. #22608684C2semidirect
9. #22608621C2semidirect
11. C73×#3104C2semidirect
22. #17988718C22semidirect
23. Q8×#10596831C22semidirect
32. #17988339C22semidirect
33. #17988337C22semidirect
34. #17988336C22semidirect
35. #17988333C22semidirect
41. #477×C73C22semidirect
42. #477×C73C22semidirect
43. #462×C73C22semidirect
44. #462×C73C22semidirect
246. #3104D73semidirect
256. C73#19174semidirect
265. #477C2×D73semidirect
266. #477C2×D73semidirect
267. #462C2×D73semidirect
268. #462C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226203072
C2×#179887872
Q8×#119240122
#226095012
#226094732
#226093772
#226093492
#226086842
#226086212
#226085982
C73×#31042
#1917473