Finite Group Databasebeta

← #33068458

Subgroups — #33068458

Order 18688.

664 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620510C211yesmaximal, characteristic, complemented
2#22619991C212yesmaximal, complemented
3#22609347C212yesmaximal, complemented
4#22609340C212yesmaximal, complemented
5#22609335C211yesmaximal, characteristic, complemented
6#22609330C211yesmaximal, characteristic, complemented
7#22609291C211yesmaximal, characteristic, complemented
8#22609286C211yesmaximal, characteristic, complemented
9#22608679C211yesmaximal, characteristic, complemented
10#22608591C212yesmaximal, complemented
11C8×#41×C73C211yesmaximal, characteristic, complemented
12#17989137C2212yes—
13#17988992C2211yescharacteristic
14#17988964C2211yescharacteristic
15#17988934C2212yes—
16#17988911—22no—
17#17988882C2212yes—
18#17988877C2212yes—
19#17988823—21no—
20#17988823—21no—
21#17988787C2212yes—
22#17988782C2212yes—
23#17988732—22no—
24C4×#11924015—22no—
25C4×D584—21no—
26C4×D584—21no—
27C4×#11924013—22no—
28#17988640C2212yes—
29#17988340C2211yescharacteristic
30#17988340C2211yescharacteristic
31#17988338C2212yes—
32#17988335C2212yes—
33#17988335C2212yes—
34#17988330C2212yes—
35#17988330C2212yes—
36C4×#41×C73C2211yescharacteristic
37C8×Q8×C73C2211yescharacteristic
38C8×Q8×C73C2211yescharacteristic
39C8×D4×C73—21no—
40C8×D4×C73—21no—
41C8×D4×C73C2211yescharacteristic
42C8×D4×C73C2211yescharacteristic
43#431×C73—21no—
44#431×C73C2211yescharacteristic
45C2×C4×C8×C73—21no—
46C2×C4×C8×C73C2211yescharacteristic
47C2×#11924028—22no—
48C2×#11924015—22no—
49C2×D584—21no—
50C2×D584—21no—
51C2×#11924013—22no—
52#14556591C2312yes—
53#14556588C2312yes—
54#14556579—22no—
55#14556565—22no—
56#14556565—22no—
57C4×D292—22no—
58C4×D292—22no—
59#14556525—21no—
60#14556525—21no—
61#14556524—22no—
62#14556522—22no—
63C4⋊#10596831C2312yes—
64C4⋊#10596831C2312yes—
65#14556515—22no—
66#14556513—22no—
67#14556513—22no—
68#14556512—22no—
69#14556511—22no—
70#14556511—22no—
71C22⋊#10596831—22no—
72C4×#10596831—22no—
73C2×#41×C73D411yescharacteristic
74C22×C8×C73—21no—
75C22×C8×C73—21no—
76C22×C8×C73D411yescharacteristic
77C22×C8×C73D411yescharacteristic
78C4×Q8×C73D411yescharacteristic
79C4×Q8×C73C2311yescharacteristic
80C4×D4×C73—21no—
81C4×D4×C73—21no—
82C4×D4×C73D411yescharacteristic
83C4×D4×C73C2311yescharacteristic
84#117×C73—21no—
85#117×C73D411yescharacteristic
86C2×C42×C73—21no—
87C2×C42×C73C2311yescharacteristic
88C4⋊C8×C73—21no—
89C4⋊C8×C73—21no—
90C4⋊C8×C73C2311yescharacteristic
91C4⋊C8×C73C2311yescharacteristic
92C22⋊C8×C73—21no—
93C22⋊C8×C73—21no—
94C22⋊C8×C73—21no—
95C4×C8×C73—21no—
96C4×C8×C73—21no—
97C4×C8×C73—21no—
98C4×C8×C73D411yescharacteristic
99C4×C8×C73D411yescharacteristic
100C4×C8×C73C2311yescharacteristic
101C4×C8×C73C2311yescharacteristic
102#11924028—84no—
103C2×Dic146—22no—
104C2×D292—22no—
105C2×D292—22no—
106C2×D292—22no—
107C2×C22⋊D73—42no—
108C2×C4×D73—42no—
109#11924015—44no—
110D584—42no—
111D584—42no—
112#11924013—44no—
113C2×#10596831—22no—
114C2×#10596831—22no—
115C4⋊Dic73—22no—
116C4⋊Dic73—22no—
117#11924005—42no—
118#11924005—42no—
119#41×C73—22no—
120#41×C73—22no—
121C2×Q8×C73C2×D411yescharacteristic
122C2×D4×C73—21no—
123C2×D4×C73C2×D411yescharacteristic
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—21no—
127C22×C4×C73C2×D411yescharacteristic
128C22×C4×C73C2×D411yescharacteristic
129C22×C4×C73#4111yescharacteristic
130C2×C8×C73—22no—
131C2×C8×C73—21no—
132C2×C8×C73—21no—
133C2×C8×C73—41no—
134C2×C8×C73—42no—
135C2×C8×C73—42no—
136C2×C8×C73C2×D411yescharacteristic
137C2×C8×C73C2×D411yescharacteristic
138C2×C8×C73C2×D411yescharacteristic
139C2×C8×C73C2×D411yescharacteristic
140C4⋊C4×C73—21no—
141C4⋊C4×C73—21no—
142C4⋊C4×C73C2×D411yescharacteristic
143C4⋊C4×C73C2×D411yescharacteristic
144C22⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C22⋊C4×C73—21no—
147C42×C73—21no—
148C42×C73—21no—
149C42×C73—21no—
150C42×C73C2×D411yescharacteristic
151C42×C73C2×D411yescharacteristic
152C42×C73#4111yescharacteristic
153C42×C73C2411yescharacteristic
154C23×C73—21no—
155C23×C73#12111yescharacteristic
156C22×D73—42no—
157Q8×C73—21no—
158Q8×C73#13512yes—
159D4×C73—24no—
160D4×C73—21no—
161D4×C73#13512yes—
162C2×C4×C73—24no—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—21no—
168C2×C4×C73—22no—
169C2×C4×C73—21no—
170C2×C4×C73—21no—
171C2×C4×C73—41no—
172C2×C4×C73—41no—
173C2×C4×C73—42no—
174C2×C4×C73#12111yescharacteristic
175C2×C4×C73#12111yescharacteristic
176C2×C4×C73#12111yescharacteristic
177C2×C4×C73C22×D411yescharacteristic
178C2×C4×C73C22×D411yescharacteristic
179C2×C4×C73C2×#4111yescharacteristic
180C22⋊D73—84no—
181C2×Dic73—42no—
182D292—42no—
183D292—42no—
184D292—42no—
185D292—42no—
186D292—82no—
187C4×D73—84no—
188Dic146—82no—
189C8×C73—21no—
190C8×C73—21no—
191C8×C73—21no—
192C8×C73—41no—
193C8×C73—41no—
194C8×C73#13512yes—
195#10596831—42no—
196#10596831—42no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—22no—
200C22×C73—21no—
201C22×C73—41no—
202C22×C73—41no—
203C22×C73C2×#12111yescharacteristic
204C2×D73—82no—
205C2×D73—82no—
206C2×D73—82no—
207C4×C73—21no—
208C4×C73—21no—
209C4×C73—22no—
210C4×C73—21no—
211C4×C73—41no—
212C4×C73—41no—
213C4×C73#47212yes—
214C4×C73C2×#13511yescharacteristic
215C4×C73C2×#13511yescharacteristic
216C4×C73C2×#13511yescharacteristic
217C4×C73#57511yescharacteristic
218Dic73—82no—
219#19128—731nomaximal, Sylow
220C2×C73—41no—
221C2×C73—42no—
222C2×C73#303511yescharacteristic
223C2×C73#305011yescharacteristic
224C2×C73C2×#47211yescharacteristic
225D73—162no—
226#3169—731no—
227#3106—732no—
228C8×#41D7311yescharacteristic, complemented
229#1678—732no—
230#1677—732no—
231#1676—731no—
232#1675—731no—
233#1668—731no—
234#1667—731no—
235#1564—731no—
236#1543—732no—
237C73#1912811yescharacteristic, complemented, Sylow
238#532—732no—
239C4×#41C2×D7311yescharacteristic
240#500—731no—
241#493—731no—
242#486—732no—
243#482—1462no—
244#477—732no—
245#476—732no—
246#466—1461no—
247#466—1461no—
248#460—732no—
249#459—732no—
250#449—1462no—
251C8×Q8C2×D7311yescharacteristic
252C8×Q8C2×D7311yescharacteristic
253C4×SD16—1462no—
254C4×D8—1462no—
255C4×D8—1461no—
256C4×D8—1461no—
257C8×D4—21no—
258C8×D4—21no—
259C8×D4C2×D7311yescharacteristic
260C8×D4C2×D7311yescharacteristic
261#431—21no—
262#431—732no—
263#431C2×D7311yescharacteristic
264C2×C4×C8—21no—
265C2×C4×C8C2×D7311yescharacteristic
266#335—731no—
267#335—731no—
268#334—732no—
269#333—732no—
270#333—732no—
271#331—732no—
272#331—732no—
273C2×#41—1462no—
274C2×#41D29211yescharacteristic
275C2×SD16—1462no—
276C2×D8—1462no—
277C2×D8—1461no—
278C2×D8—1461no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8C22⋊D7311yescharacteristic
282C22×C8C22⋊D7311yescharacteristic
283C4⋊Q8—732no—
284#127—732no—
285#124—1462no—
286#121—1462no—
287#121—1462no—
288C4×Q8D29211yescharacteristic
289C4×Q8C22×D7311yescharacteristic
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—1462no—
293C4×D4—1462no—
294C4×D4D29211yescharacteristic
295C4×D4C22×D7311yescharacteristic
296#117—21no—
297#117D29211yescharacteristic
298C2×C42—21no—
299C2×C42C22×D7311yescharacteristic
300#107—1461no—
301#107—1461no—
302#107—1462no—
303#106—1462no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—732no—
307C4⋊C8—732no—
308C4⋊C8C22×D7311yescharacteristic
309C4⋊C8C22×D7311yescharacteristic
310#103—1462no—
311#102—1462no—
312#102—1462no—
313#102—1462no—
314#102—1462no—
315#102—1462no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—1462no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—1462no—
324C4×C8C22⋊D7311yescharacteristic
325C4×C8C22⋊D7311yescharacteristic
326C4×C8C22×D7311yescharacteristic
327C4×C8C22×D7311yescharacteristic
328#41—22no—
329#41—22no—
330#41—5844no—
331C2×Q8—1462no—
332C2×Q8C2×D29211yescharacteristic
333C2×D4—21no—
334C2×D4—2922no—
335C2×D4—1462no—
336C2×D4—1462no—
337C2×D4—1462no—
338C2×D4C2×D29211yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—2922no—
343C22×C4C2×C22⋊D7311yescharacteristic
344C22×C4C2×D29211yescharacteristic
345C22×C4#1192402811yescharacteristic
346SD16—2924no—
347D8—2922no—
348D8—2924no—
349D8—2922no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—1462no—
357C2×C8—1462no—
358C2×C8C2×C22⋊D7311yescharacteristic
359C2×C8C2×C22⋊D7311yescharacteristic
360C2×C8C2×C22⋊D7311yescharacteristic
361C2×C8C2×C22⋊D7311yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—1462no—
365C4⋊C4—1462no—
366C4⋊C4C2×D29211yescharacteristic
367C4⋊C4C2×D29211yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—2922no—
372C22⋊C4—2922no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42C2×C22⋊D7311yescharacteristic
377C42C2×D29211yescharacteristic
378C42#1192402811yescharacteristic
379C42C23×D7311yescharacteristic
380C23—21no—
381C23—2922no—
382C23#1455656511yescharacteristic
383Q8—21no—
384Q8—5842no—
385Q8#1455661312yes—
386D4—24no—
387D4—21no—
388D4—2922no—
389D4—2922no—
390D4—2922no—
391D4—2922no—
392D4—5842no—
393D4—5844no—
394D4#1455661312yes—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—2922no—
408C2×C4—5844no—
409C2×C4#1455656511yescharacteristic
410C2×C4#1455656511yescharacteristic
411C2×C4#1455656511yescharacteristic
412C2×C4C22×C22⋊D7311yescharacteristic
413C2×C4C22×D29211yescharacteristic
414C2×C4C2×#1192402811yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8—41no—
419C8—41no—
420C8—2922no—
421C8—2922no—
422C8#1455661112yes—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—5842no—
430C22—5842no—
431C22—5842no—
432C22C2×#1455656511yescharacteristic
433C4—21no—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—5842no—
440C4#1798885712yes—
441C4C2×#1455661111yescharacteristic
442C4C2×#1455661311yescharacteristic
443C4C2×#1455661311yescharacteristic
444C4#1798938511yescharacteristic
445C2—41no—
446C2—42no—
447C2—11682no—
448C2#2261938611yescharacteristic
449C2#2261951211yescharacteristic
450C2C2×#1798885711yescharacteristic
subgroupG/H
centerC2×C4#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC8×#41×C73C2
socleC22×C73C2×#121
hypercenterC8×#41D73
hypocenterC73#19128
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. #22620510C2semidirect
2. #22619991C2semidirect
3. #22609347C2semidirect
4. #22609340C2semidirect
5. #22609335C2semidirect
6. #22609330C2semidirect
7. #22609291C2semidirect
8. #22609286C2semidirect
9. #22608679C2semidirect
10. #22608591C2semidirect
11. C8×#41×C73C2semidirect
228. C8×#41D73semidirect
237. C73#19128semidirect

Maximal subgroups

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

subgroupindex
#226205102
#226199912
#226093472
#226093402
#226093352
#226093302
#226092912
#226092862
#226086792
#226085912
C8×#41×C732
#1912873