Finite Group Databasebeta

← #10691750

Subgroups — #10691750

Order 768.

522 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82741C211yesmaximal, characteristic
2#73339C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic
5#69946C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9276—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13#5228C412yescomplemented
14#5212C2211yescharacteristic
15#5210—21no—
16C4×#217—21no—
17C4×#217C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5051—22no—
20#5050—21no—
21#5044—21no—
22#4970C412yescomplemented
23#4653—21no—
24#4652—21no—
25#4649—21no—
26#4646—21no—
27C3×#328—21no—
28C3×#328—21no—
29#4526—21no—
30#4526—21no—
31#4467—22no—
32#3189—31no—
33#2368—31no—
34#1996—31no—
35#1520—31no—
36C2×#328D311yescharacteristic
37#1294—31no—
38#1282—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×Dic6D411yescharacteristic
43C22×C3×C8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217D411yescharacteristic
54C2×#217D411yescharacteristic
55C2×#216—22no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D411yescharacteristic
58#924—21no—
59#924—21no—
60#924C2×C412yes—
61#924D412yes—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C22×C3⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72C3×#95D411yescharacteristic
73C3×#95D411yescharacteristic
74#835—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830C2×C412yes—
83#810—22no—
84#807—22no—
85C2×C22⋊Q8—31no—
86C2×C22⋊Q8C2×D311yescharacteristic
87#484—32no—
88#484—61no—
89#467—62no—
90#451—32no—
91#451—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8C2×D311yescharacteristic
94#391—61no—
95#387—61no—
96#387—61no—
97#385—61no—
98#380—31no—
99C4×C4⋊C4—61no—
100C4×C22⋊C4—31no—
101C4×C22⋊C4—61no—
102C2×#95C2×D311yescharacteristic
103#342—61no—
104#342—61no—
105#328—21no—
106#328—21no—
107#328—62no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—81no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—42no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—81no—
120C22×C3×C4—81no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C2×Dic6—21no—
133C2×Dic6—81no—
134C2×Dic6C22⋊C412yes—
135C2×C3×C8—41no—
136C2×C3×C8—81no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143#217—22no—
144#217—22no—
145#217—21no—
146#217—21no—
147#217—21no—
148#217—41no—
149#217—41no—
150#217Q1612yescomplemented
151#216—82no—
152C4⋊Dic3—21no—
153C4⋊Dic3—81no—
154C4⋊Dic3C22⋊C412yes—
155Dic3⋊C4—22no—
156Dic3⋊C4—22no—
157Dic3⋊C4—42no—
158Dic3⋊C4—81no—
159C4×Dic3—42no—
160C4×Dic3—81no—
161C4×Dic3—81no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—81no—
164C22×Q8—31no—
165C22×Q8C4×D311yescharacteristic
166C23×C4—21no—
167C23×C4C22×D311yescharacteristic
168C2×Q16—62no—
169C22×C8—21no—
170C22×C8—61no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—32no—
176C22⋊Q8—32no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4C4×D311yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C42—61no—
192C2×C42—61no—
193C2×C42—61no—
194#107—62no—
195#103—62no—
196#103—62no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199C22⋊C8—32no—
200C22⋊C8—61no—
201#95—21no—
202#95—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208#95#6711yescharacteristic
209#95#6711yescharacteristic
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C23×C3C2×Q1611yescharacteristic
216C23×C3#13511yescharacteristic
217C23×C3#13711yescharacteristic
218C23×C3#13711yescharacteristic
219C3×Q8—42no—
220C3×Q8—81no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—82no—
236C2×C3×C4—82no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258Dic6—42no—
259Dic6—81no—
260C3×C8—81no—
261C3⋊C8—42no—
262C24D3×D411yescharacteristic
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—32no—
266C2×Q8—61no—
267C2×Q8—81no—
268C2×Q8—241no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—121no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4#23711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289Q16—242no—
290C2×C8—41no—
291C2×C8—62no—
292C2×C8—81no—
293C2×C8—241no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—32no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—61no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—122no—
303C4⋊C4—241no—
304C4⋊C4—241no—
305C22⋊C4—32no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C42—122no—
316C42—241no—
317C42—241no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3#38611yescharacteristic
329C22×C3C4×Q1611yescharacteristic
330C22×C3#44111yescharacteristic
331C22×C3#45111yescharacteristic
332C22×C3#45211yescharacteristic
333C22×C3#46211yescharacteristic
334C22×C3#46411yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—42no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4—81no—
343Dic3—42no—
344Dic3—42no—
345Dic3—42no—
346Dic3—81no—
347Dic3—161no—
348C23—41no—
349C23—41no—
350C23#67⋊C411yescharacteristic
351C23#88411yescharacteristic
352C23#91511yescharacteristic
353C23#91611yescharacteristic
354C23D3×Q1611yescharacteristic
355C23#91811yescharacteristic
356C23#93711yescharacteristic
357Q8—42no—
358Q8—81no—
359Q8—122no—
360Q8—241no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4D3×C22⋊C411yescharacteristic
395C2×C4#88311yescharacteristic
396C2×C4C4×#6711yescharacteristic
397C2×C4#93811yescharacteristic
398C8—81no—
399C8—122no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3#187811yescharacteristic
407C2×C3#190611yescharacteristic
408C2×C3#191311yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#478911yescharacteristic
420C22#479011yescharacteristic
421C22#479511yescharacteristic
422C22#479811yescharacteristic
423C22#495611yescharacteristic
424C22#517311yescharacteristic
425C22#518611yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—122no—
435C4—122no—
436C4—122no—
437C4—241no—
438C4—481no—
439C3#927611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#7003411yescharacteristic
447C2#7028111yescharacteristic
448C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9276
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
13. #5228C4semidirect
22. #4970C4semidirect
150. #217Q16semidirect
439. C3#9276semidirect

Maximal subgroups

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

subgroupindex
#827412
#733392
C2×C3×#3282
#717012
#699462
#676772
#676122
#92763