Finite Group Databasebeta

← #10692828

Subgroups — #10692828

Order 768.

474 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 89 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#82774C211yesmaximal, characteristic
2#78620C211yesmaximal, characteristic
3C3×#1511C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#69534C211yesmaximal, characteristic
6#67682C211yesmaximal, characteristic
7#67628C211yesmaximal, characteristic
8#9444—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5229C411yescharacteristic, complemented
14#5229C411yescharacteristic, complemented
15#5211C2211yescharacteristic
16C4×#217C2211yescharacteristic
17#5207C2211yescharacteristic
18#5118—21no—
19Dic3×M16—21no—
20#4978—21no—
21#4978—21no—
22#4977C411yescharacteristic, complemented
23#4975C411yescharacteristic, complemented
24#4952—21no—
25#4952—21no—
26#4720—22no—
27#4720—22no—
28C3×#328—22no—
29#4538—22no—
30#4527—21no—
31#4527—21no—
32#2921—31no—
33#2499—31no—
34#1820—31no—
35#1511—31no—
36#1511D311yescharacteristic
37#1284—31no—
38#1284—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C2×C3×M16—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217D411yescharacteristic
53C2×#217D411yescharacteristic
54#926—21no—
55#926—21no—
56#926—21no—
57#926—21no—
58C2×C4⋊Dic3C2×C411yescharacteristic
59C2×C4⋊Dic3C2×C411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3C2×C411yescharacteristic
63C2×C4×Dic3C2×C411yescharacteristic
64C2×C4×Dic3D411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C2×#208—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8D412yes—
69C3×#95—21no—
70C3×#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830D412yes—
82#815—22no—
83Dic3⋊C8—24no—
84C8×Dic3—22no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C2×#117—31no—
87#433—62no—
88#433—62no—
89#423—61no—
90#407—31no—
91#407C2×D311yescharacteristic
92C4×M16—61no—
93#397—61no—
94#397—61no—
95#391—31no—
96#391—31no—
97#387—61no—
98#387—61no—
99#384—31no—
100#382—31no—
101#381—31no—
102#381C2×D311yescharacteristic
103C4×C22⋊C4—31no—
104#342—61no—
105#342—61no—
106#328—22no—
107#328—62no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—41no—
127C22×Dic3C22⋊C411yescharacteristic
128C22×Dic3C22⋊C411yescharacteristic
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142#217—21no—
143#217—21no—
144#217—21no—
145#217—21no—
146#217—42no—
147#217—41no—
148#217—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—42no—
154Dic3⋊C4—42no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—22no—
164C22×Q8C4×D311yescharacteristic
165C23×C4C22×D311yescharacteristic
166C2×M16—21no—
167C2×M16—61no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172#117—61no—
173#117—61no—
174#117—61no—
175#117—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4C4×D311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—31no—
189C2×C42—31no—
190C2×C42—31no—
191C2×C42—31no—
192C4⋊C8—64no—
193C22⋊C8—21no—
194C22⋊C8—32no—
195C22⋊C8—61no—
196C22⋊C8#6712yes—
197#97—62no—
198C4×C8—62no—
199#95—21no—
200#95—21no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208#95—61no—
209#95—61no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3C2×C22⋊C411yescharacteristic
215C23×C3#12111yescharacteristic
216C23×C3#12111yescharacteristic
217C3×Q8—81no—
218C3×Q8—81no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
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—81no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12311yescharacteristic
240C2×Dic3—22no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—41no—
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—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C3×C8—42no—
258C3⋊C8—42no—
259C24D3×D411yescharacteristic
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—41no—
263C2×Q8—41no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4#23511yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282M16—41no—
283M16—41no—
284M16—121no—
285M16—121no—
286C2×C8—22no—
287C2×C8—62no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—122no—
297C4⋊C4—122no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—122no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C42—61no—
310C42—61no—
311C42—61no—
312C42—61no—
313C42—121no—
314C42—121no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#38611yescharacteristic
326C22×C3#41011yescharacteristic
327C22×C3#41011yescharacteristic
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4—161no—
335Dic3—42no—
336Dic3—41no—
337Dic3—41no—
338Dic3—41no—
339Dic3—41no—
340Dic3—81no—
341Dic3—161no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23D3×C22⋊C411yescharacteristic
347C23#88411yescharacteristic
348C23#93811yescharacteristic
349Q8—81no—
350Q8—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—62no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4#67⋊C411yescharacteristic
386C2×C4#88311yescharacteristic
387C2×C4C4×#6711yescharacteristic
388C2×C4#93711yescharacteristic
389C8—42no—
390C8—122no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3#188411yescharacteristic
396C2×C3#189911yescharacteristic
397C2×C3#190911yescharacteristic
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—81no—
407C22—81no—
408C22#474011yescharacteristic
409C22#474011yescharacteristic
410C22#495611yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—81no—
416C4—81no—
417C4—122no—
418C4—121no—
419C4—121no—
420C4—121no—
421C4—121no—
422C4—161no—
423C4—241no—
424C4—481no—
425C3#944411yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—41no—
429C2—41no—
430C2#7002911yescharacteristic
431C2#7035011yescharacteristic
432C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1511C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1511D3
hypocenterC3#9444
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
13. #5229C4semidirect
14. #5229C4semidirect
22. #4977C4semidirect
23. #4975C4semidirect
425. C3#9444semidirect

Maximal subgroups

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

subgroupindex
#827742
#786202
C3×#15112
#716892
#695342
#676822
#676282
#94443