Finite Group Databasebeta

← #10691154

Subgroups — #10691154

Order 768.

530 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 91 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#78634C211yesmaximal, characteristic
2#74091C211yesmaximal, characteristic
3#69980C211yesmaximal, characteristic, complemented
4C4×#806C211yesmaximal, characteristic
5#69445C211yesmaximal, characteristic, complemented
6C3×#1284C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9184—31nomaximal, Sylow
9C2×C4×#67C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16#5128—22no—
17#4961—21no—
18C2×#806—21no—
19#4953—21no—
20#4729—22no—
21#4726—21no—
22#4718—21no—
23#4674C411yescharacteristic, complemented
24#4673C411yescharacteristic, complemented
25#4668—21no—
26#4661C411yescharacteristic, complemented
27#4656C411yescharacteristic, complemented
28#4653—21no—
29#4527—21no—
30#4467—22no—
31#4452—22no—
32#2975—31no—
33#2638—31no—
34C4×#99—31no—
35#1508—31no—
36#1508—31no—
37#1284—31no—
38#1284D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C42—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217D411yescharacteristic
50C4×#67—21no—
51C4×#67—21no—
52C4×#67—21no—
53C4×#67—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4C2×C411yescharacteristic
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3C2×C411yescharacteristic
60#924—21no—
61#924—21no—
62#924—21no—
63#924—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4C2×C411yescharacteristic
66C2×C4×Dic3—21no—
67C2×C4×Dic3D411yescharacteristic
68C2×#208—21no—
69C2×ΓL(1,𝔽25)—22no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95C2×C411yescharacteristic
75C3×#95C2×C411yescharacteristic
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830D412yes—
83(C2×D3)⋊C8—22no—
84Dic3⋊C8—22no—
85C8×Dic3—22no—
86#806—24no—
87C2×C22⋊Q8—31no—
88C2×C4×D4—31no—
89#436—62no—
90#436—62no—
91#431—61no—
92C2×#99—61no—
93#407—31no—
94#407—61no—
95#407C2×D311yescharacteristic
96#398—31no—
97#397—31no—
98#397—61no—
99#395—31no—
100#394—61no—
101#393—31no—
102#387—61no—
103#385—61no—
104#381C2×D311yescharacteristic
105C4×C22⋊C4—31no—
106C4×C22⋊C4C2×D311yescharacteristic
107#342—61no—
108#328—62no—
109#328—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—81no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—21no—
123C2×#67—21no—
124C2×#67—22no—
125C2×#67—41no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—41no—
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C22×Dic3C22⋊C411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—81no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—81no—
138C3×M16—81no—
139C2×C3×C8—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—81no—
143C3×C42—81no—
144#217—21no—
145#217—21no—
146#217—22no—
147#217—41no—
148#217—41no—
149#217—41no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—42no—
153(C2×D3)⋊C4—81no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—42no—
157C4⋊Dic3—81no—
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—81no—
165#208—81no—
166C2×C3⋊C8—22no—
167ΓL(1,𝔽25)—82no—
168C22×Q8—31no—
169C22×D4—31no—
170C23×C4—31no—
171C23×C4C22×D311yescharacteristic
172C2×M16—21no—
173C2×M16—62no—
174C2×M16—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C22⋊Q8—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C42—21no—
197C2×C42—31no—
198C2×C42—61no—
199C4⋊C8—62no—
200#99—64no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—62no—
204C22⋊C8—61no—
205C22⋊C8D1212yes—
206C4×C8—62no—
207#95—21no—
208#95—21no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95C4×D311yescharacteristic
215#95C4×D311yescharacteristic
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—41no—
227C22×D3—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—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12311yescharacteristic
247#67—42no—
248#67—42no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—81no—
263C2×Dic3—81no—
264C2×Dic3#10412yes—
265C4×D3—42no—
266C4×D3—81no—
267Dic6—81no—
268Dic6—81no—
269C3×C8—42no—
270C3⋊C8—42no—
271C24—31no—
272C24D3×D411yescharacteristic
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—241no—
276C2×D4—62no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—121no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—81no—
291C22×C4—121no—
292C22×C4—241no—
293C22×C4C2×C4×D311yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4#23911yescharacteristic
299M16—81no—
300M16—242no—
301M16—241no—
302C2×C8—22no—
303C2×C8—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—122no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C4⋊C4—241no—
312C22⋊C4—22no—
313C22⋊C4—41no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—62no—
319C22⋊C4—81no—
320C22⋊C4—122no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—241no—
325C42—61no—
326C42—61no—
327C42—81no—
328C42—241no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3#38611yescharacteristic
340C22×C3C2×#10411yescharacteristic
341C22×C3#42111yescharacteristic
342C2×D3—42no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—81no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—81no—
352C3×C4—81no—
353C3×C4—82no—
354Dic3—22no—
355Dic3—41no—
356Dic3—41no—
357Dic3—82no—
358Dic3—81no—
359Dic3—81no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—121no—
368C23—121no—
369C23D3×C22⋊C411yescharacteristic
370C23#88411yescharacteristic
371C23#89511yescharacteristic
372Q8—241no—
373Q8—241no—
374D4—122no—
375D4—122no—
376C2×C4—32no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4C4×D1211yescharacteristic
410C2×C4#88211yescharacteristic
411C2×C4#89311yescharacteristic
412C2×C4#89411yescharacteristic
413C8—42no—
414C8—122no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3#187111yescharacteristic
420C2×C3#190911yescharacteristic
421C2×C3#191311yescharacteristic
422D3—81no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—81no—
432C22—81no—
433C22—122no—
434C22—121no—
435C22—121no—
436C22—121no—
437C22—241no—
438C22#466511yescharacteristic
439C22D3×#10411yescharacteristic
440C22#482111yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—62no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—121no—
450C4—121no—
451C4—242no—
452C4—241no—
453C4—241no—
454C3#918411yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—241no—
460C2#6871211yescharacteristic
461C2#6875511yescharacteristic
462C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9184
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
3. #69980C2semidirect
5. #69445C2semidirect
6. C3×#1284C2semidirect
7. #67628C2semidirect
23. #4674C4semidirect
24. #4673C4semidirect
26. #4661C4semidirect
27. #4656C4semidirect
38. #1284D3semidirect
454. C3#9184semidirect

Maximal subgroups

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

subgroupindex
#786342
#740912
#699802
C4×#8062
#694452
C3×#12842
#676282
#91843