Finite Group Databasebeta

← #10692938

Subgroups — #10692938

Order 768.

726 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 97 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#82654C211yesmaximal, characteristic
2#74053C211yesmaximal, characteristic, complemented
3C3×#1510C211yesmaximal, characteristic
4#71674C211yesmaximal, characteristic
5#69990C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9466—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#936C412yescomplemented
12C2×#877—21no—
13C3×#407C2211yescharacteristic
14C3×#381C2211yescharacteristic
15#5220—21no—
16#5220—21no—
17#5211C2211yescharacteristic
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20C2×#824—22no—
21#4973C412yescomplemented
22#4947—22no—
23#4943—21no—
24#4646—22no—
25#4646—22no—
26C3×#327—22no—
27#4537—22no—
28#2436—31no—
29#2310—31no—
30#1784—31no—
31#1510—31no—
32#1510D311yescharacteristic
33#1284—31no—
34#1284—31no—
35C22×C3×D4—21no—
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C23×Dic3D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—44no—
45C22×Dic6C2×C412yes—
46C22×Dic6D412yes—
47C2×C3×M16—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—22no—
58C2×#217—22no—
59#936—44no—
60C2×Dic3⋊C4—22no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3C2×C412yes—
63C2×#208—21no—
64#877—44no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—24no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#824—44no—
78C23×Q8—31no—
79C2×#124—32no—
80C2×#121C2×D311yescharacteristic
81C2×#117—61no—
82C2×#101—62no—
83#407—31no—
84#407C2×D311yescharacteristic
85#393—62no—
86#391—32no—
87#391—62no—
88#391—62no—
89#386—61no—
90#386—61no—
91#381—31no—
92#381—61no—
93#381C2×D311yescharacteristic
94C4×C22⋊C4—31no—
95#327—22no—
96#327—62no—
97C24×C3—21no—
98C24×C3C2×D411yescharacteristic
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—42no—
102C2×C3×D4—44no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—42no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—24no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—42no—
122C22×Dic3C22⋊C412yes—
123C22×Dic3C22⋊C412yes—
124C2×Dic6—22no—
125C2×Dic6—22no—
126C2×Dic6—22no—
127C2×Dic6—22no—
128C2×Dic6—24no—
129C2×Dic6—24no—
130C2×Dic6—42no—
131C2×Dic6—42no—
132C2×Dic6—42no—
133C2×Dic6—42no—
134C2×Dic6—42no—
135C2×Dic6—42no—
136C2×Dic6—44no—
137C2×Dic6—44no—
138C3×M16—42no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145#217—44no—
146#217—42no—
147#217—42no—
148#217—44no—
149Dic3⋊C4—44no—
150C4×Dic3—44no—
151C4×Dic3—42no—
152#208—42no—
153C2×C3⋊C8—41no—
154C22×Q8—32no—
155C22×Q8—32no—
156C22×Q8—61no—
157C22×Q8—61no—
158C22×Q8—61no—
159C22×Q8—61no—
160C22×Q8—124no—
161C22×D4—21no—
162C22×D4C4×D311yescharacteristic
163C23×C4—31no—
164C23×C4—31no—
165C23×C4C22×D311yescharacteristic
166C2×M16—21no—
167C2×M16—61no—
168#124—124no—
169#121—21no—
170#121—21no—
171#121—21no—
172#121—21no—
173#117—124no—
174C2×C4⋊C4—62no—
175C2×C4⋊C4C4×D311yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C42—32no—
183C2×C42—62no—
184#101—124no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#95—21no—
190#95—32no—
191#95—62no—
192#95—62no—
193#95—64no—
194#95—61no—
195#95—61no—
196#95#6712yes—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3C2×C22⋊C411yescharacteristic
205C23×C3#12011yescharacteristic
206C23×C3#12311yescharacteristic
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—82no—
210C3×D4—82no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—24no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3#9914yes—
244Dic6—44no—
245Dic6—44no—
246Dic6—44no—
247Dic6—44no—
248C3×C8—81no—
249C3⋊C8—81no—
250C24—21no—
251C24D3×D411yescharacteristic
252C2×Q8—64no—
253C2×Q8—64no—
254C2×Q8—62no—
255C2×Q8—62no—
256C2×Q8—62no—
257C2×Q8—62no—
258C2×Q8—122no—
259C2×Q8—122no—
260C2×Q8—124no—
261C2×Q8—122no—
262C2×Q8—122no—
263C2×Q8—124no—
264C2×Q8—122no—
265C2×Q8—122no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—42no—
269C2×D4—44no—
270C2×D4—41no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—32no—
275C22×C4—32no—
276C22×C4—42no—
277C22×C4—64no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—122no—
285C22×C4C2×C4×D311yescharacteristic
286C22×C4#23511yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4#23711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291M16—42no—
292M16—122no—
293C2×C8—41no—
294C2×C8—121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—124no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—124no—
302C22⋊C4—124no—
303C22⋊C4—122no—
304C22⋊C4—122no—
305C42—122no—
306C42—124no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—44no—
315C22×C3—84no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#38611yescharacteristic
320C22×C3C2×#9912yes—
321C22×C3#45412yes—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—82no—
327C3×C4—81no—
328Dic3—24no—
329Dic3—42no—
330Dic3—82no—
331Dic3—82no—
332C23—22no—
333C23—22no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—42no—
339C23D3×C22⋊C411yescharacteristic
340C23#88311yescharacteristic
341C23#93711yescharacteristic
342Q8—124no—
343Q8—124no—
344Q8—124no—
345Q8—124no—
346D4—81no—
347D4—81no—
348D4—82no—
349D4—82no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—34no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—64no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4#67⋊C411yescharacteristic
380C2×C4#88411yescharacteristic
381C2×C4C4×#6711yescharacteristic
382C2×C4#93811yescharacteristic
383C8—81no—
384C8—241no—
385C2×C3—22no—
386C2×C3—22no—
387C2×C3—41no—
388C2×C3—161no—
389C2×C3#188811yescharacteristic
390C2×C3#189811yescharacteristic
391C2×C3#191211yescharacteristic
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22D3×#9912yes—
405C22#495611yescharacteristic
406C22#520112yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—64no—
412C4—82no—
413C4—81no—
414C4—122no—
415C4—242no—
416C4—242no—
417C3#946611yescharacteristic, complemented, Sylow
418C2—22no—
419C2—22no—
420C2—41no—
421C2—161no—
422C2#7003611yescharacteristic
423C2#7032511yescharacteristic
424C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1510D3
hypocenterC3#9466
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #74053C2semidirect
5. #69990C2semidirect
6. #67682C2semidirect
7. #67628C2semidirect
11. C2×#936C4semidirect
21. #4973C4semidirect
417. C3#9466semidirect

Maximal subgroups

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

subgroupindex
#826542
#740532
C3×#15102
#716742
#699902
#676822
#676282
#94663