Finite Group Databasebeta

← #10692142

Subgroups — #10692142

Order 768.

464 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 85 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#82628C211yesmaximal, characteristic
2#78636C211yesmaximal, characteristic, complemented
3C3×#1510C211yesmaximal, characteristic, complemented
4#71676C211yesmaximal, characteristic, complemented
5#69532C211yesmaximal, characteristic, complemented
6#67687C211yesmaximal, characteristic, complemented
7#67629C211yesmaximal, characteristic, complemented
8#9332—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10#5791C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5218C411yescharacteristic, complemented
14#5217C411yescharacteristic, complemented
15#5207C2211yescharacteristic
16#5207C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5128—21no—
19#5127—21no—
20#5042—21no—
21#5039—21no—
22#5037—21no—
23#5035—21no—
24#4971C411yescharacteristic, complemented
25#4967C411yescharacteristic, complemented
26#4729—21no—
27#4728—21no—
28#4727—21no—
29#4720—21no—
30C3×#327—21no—
31C3×#327—21no—
32#4525—21no—
33#4525—21no—
34#4452—21no—
35#4451—21no—
36#3194—31no—
37#2978—31no—
38#1516—31no—
39#1510D311yescharacteristic, complemented
40#1471—31no—
41#1306—31no—
42#1293—31no—
43C22×C3×D4—21no—
44C22×C3×D4C2×C411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C2×#235D411yescharacteristic
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—
55#953—21no—
56C2×#214—21no—
57C2×#213—21no—
58#930C2×C411yescharacteristic
59#930D411yescharacteristic
60#929—21no—
61C4×#67—21no—
62C4×#67—21no—
63#926D411yescharacteristic
64#924C2×C411yescharacteristic
65#924D411yescharacteristic
66C2×#208—21no—
67C2×#208—21no—
68C22×C3⋊C8—21no—
69C2×ΓL(1,𝔽25)—21no—
70C2×D3×C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D411yescharacteristic
73C3×C22⋊C8D411yescharacteristic
74C3×#95—21no—
75C3×#95—21no—
76#832—21no—
77#832—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830C2×C411yescharacteristic
83#830C2×C411yescharacteristic
84#830D411yescharacteristic
85#830D411yescharacteristic
86(C2×D3)⋊C8—21no—
87(C2×D3)⋊C8—21no—
88#815—21no—
89Dic3⋊C8—21no—
90Dic3⋊C8—21no—
91C8×Dic3—21no—
92#810—21no—
93#809—21no—
94#808—21no—
95#807—21no—
96#525—31no—
97C2×#121C2×D311yescharacteristic
98#483—31no—
99#482—61no—
100#480—31no—
101#469—61no—
102#468—61no—
103#450—31no—
104#449—61no—
105#447—31no—
106#436—61no—
107#436—61no—
108#435—61no—
109#435—61no—
110#433—61no—
111#408—61no—
112#407—31no—
113#407—31no—
114#407C2×D311yescharacteristic
115C2×C22⋊C8—31no—
116#381C2×D311yescharacteristic
117#328—61no—
118#327—21no—
119#327—21no—
120#327—61no—
121#323—61no—
122#323—61no—
123C24×C3—21no—
124C24×C3#4111yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—81no—
130C2×C3×D4—82no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4—81no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—21no—
141C2×#67—21no—
142C2×#67—21no—
143#235—81no—
144C2×D12C22⋊C411yescharacteristic
145C2×C4×D3—21no—
146C2×Dic6C22⋊C411yescharacteristic
147C3×M16—81no—
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—81no—
155#217—21no—
156#217—21no—
157#217—21no—
158#214—81no—
159#213—81no—
160(C2×D3)⋊C4—21no—
161(C2×D3)⋊C4—21no—
162C4⋊Dic3C22⋊C411yescharacteristic
163C4⋊Dic3C22⋊C411yescharacteristic
164Dic3⋊C4—21no—
165Dic3⋊C4—21no—
166C4×Dic3—21no—
167#208—81no—
168#208—81no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—81no—
175ΓL(1,𝔽25)—81no—
176D3×C8—81no—
177C2×#41—31no—
178C22×D4—21no—
179C22×D4C4×D311yescharacteristic
180C23×C4C22×D311yescharacteristic
181C2×SD16—61no—
182C2×D8—61no—
183C2×M16—21no—
184C2×M16—61no—
185C2×M16—61no—
186C2×M16—61no—
187C22×C8—61no—
188C22×C8—61no—
189#123—61no—
190C22⋊Q8—31no—
191C22⋊Q8—31no—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121—31no—
197#121—31no—
198#120—61no—
199C4×D4—61no—
200C4×D4—61no—
201#117—31no—
202C2×C4⋊C4C4×D311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205#107—61no—
206#106—61no—
207C4⋊C8—61no—
208C4⋊C8—61no—
209#103—61no—
210#102—61no—
211#102—61no—
212#102—61no—
213C22⋊C8—21no—
214C22⋊C8—31no—
215C22⋊C8—31no—
216C22⋊C8—31no—
217C22⋊C8—31no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8#6711yescharacteristic
225C22⋊C8#6711yescharacteristic
226#97—61no—
227C4×C8—61no—
228#95—21no—
229#95—21no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3C4×D411yescharacteristic
238C23×C3#12111yescharacteristic
239C23×C3#12311yescharacteristic
240C22×D3—21no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—81no—
244C3×D4—82no—
245C3×D4—82no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4C2×C22⋊C411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12111yescharacteristic
261#67—81no—
262#67—81no—
263#67—81no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267D12—41no—
268C4×D3—41no—
269C4×D3—41no—
270Dic6—41no—
271C3×C8—41no—
272C3×C8—41no—
273C3⋊C8—41no—
274C3⋊C8—41no—
275C3⋊C8—41no—
276C3⋊C8—41no—
277C3⋊C8—81no—
278C24—21no—
279C24#23711yescharacteristic
280#41—241no—
281C2×Q8—31no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—31no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—81no—
291C2×D4—82no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—61no—
295C22×C4—81no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4#23511yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4C2×#6711yescharacteristic
302SD16—241no—
303D8—241no—
304M16—81no—
305M16—241no—
306M16—241no—
307M16—241no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—121no—
315C2×C8—241no—
316C2×C8—241no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C22⋊C4—61no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—81no—
331C42—61no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—82no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—162no—
346C22×C3#38611yescharacteristic
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—81no—
355C3×C4—161no—
356Dic3—41no—
357Dic3—41no—
358Dic3—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—41no—
364C23—61no—
365C23—81no—
366C23—81no—
367C23#67⋊C411yescharacteristic
368C23#88311yescharacteristic
369C23#93811yescharacteristic
370Q8—121no—
371D4—41no—
372D4—41no—
373D4—81no—
374D4—82no—
375D4—82no—
376D4—121no—
377D4—241no—
378D4—241no—
379D4—241no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4D3×C22⋊C411yescharacteristic
397C2×C4#88411yescharacteristic
398C2×C4C4×#6711yescharacteristic
399C2×C4#93711yescharacteristic
400C8—41no—
401C8—41no—
402C8—121no—
403C8—121no—
404C8—121no—
405C8—121no—
406C8—241no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—161no—
411C2×C3#188111yescharacteristic
412C2×C3#190311yescharacteristic
413C2×C3#191311yescharacteristic
414D3—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—121no—
429C22—121no—
430C22—121no—
431C22—162no—
432C22#495611yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—81no—
438C4—121no—
439C4—121no—
440C4—121no—
441C4—161no—
442C3#933211yescharacteristic, complemented, Sylow
443C2—41no—
444C2—41no—
445C2—41no—
446C2—161no—
447C2—241no—
448C2#7003411yescharacteristic
449C2#7032911yescharacteristic
450C2#7041611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#386
prime radical#82628C2
cosocleC23×C3×C4C23
hypercenter#1510D3
hypocenterC3#9332
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. #78636C2semidirect
3. C3×#1510C2semidirect
4. #71676C2semidirect
5. #69532C2semidirect
6. #67687C2semidirect
7. #67629C2semidirect
13. #5218C4semidirect
14. #5217C4semidirect
24. #4971C4semidirect
25. #4967C4semidirect
39. #1510D3semidirect
442. C3#9332semidirect

Maximal subgroups

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

subgroupindex
#826282
#786362
C3×#15102
#716762
#695322
#676872
#676292
#93323