Finite Group Databasebeta

← #10692258

Subgroups — #10692258

Order 768.

470 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 93 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#78624C211yesmaximal, characteristic
2#78584C211yesmaximal, characteristic, complemented
3#69944C211yesmaximal, characteristic
4#69926C211yesmaximal, characteristic, complemented
5#69921C211yesmaximal, characteristic
6C3×#1298C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9348—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14#5212C2211yescharacteristic
15#5207C2211yescharacteristic
16#5131C411yescharacteristic, complemented
17#5131C411yescharacteristic, complemented
18#5121C411yescharacteristic, complemented
19#5121C411yescharacteristic, complemented
20#5102—21no—
21#5101—21no—
22#4737—21no—
23#4737—21no—
24#4736—21no—
25#4735—21no—
26#4720—22no—
27#4719—21no—
28#4718—21no—
29#4551—21no—
30#4551—21no—
31#4470—21no—
32#4470—21no—
33#4468—22no—
34#3094—31no—
35#2924—31no—
36#1825—31no—
37#1519—31no—
38#1471—31no—
39#1305—31no—
40#1298D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C2×C3×M16C2×C411yescharacteristic
45C2×C3×M16C2×C411yescharacteristic
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C2×#217—21no—
49C2×#217—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—21no—
55#926—21no—
56#926—21no—
57#926C2×C411yescharacteristic
58#926C2×C411yescharacteristic
59#926D411yescharacteristic
60#926D411yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4×Dic3—21no—
66C2×#208—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D411yescharacteristic
73C3×C22⋊C8D411yescharacteristic
74C3⋊#95—21no—
75#830—21no—
76#830D412yes—
77#821—21no—
78#821—21no—
79#821—21no—
80#821—21no—
81#818—21no—
82#818—21no—
83#817—21no—
84#817—21no—
85#815—22no—
86Dic3⋊C8—22no—
87Dic3⋊C8—22no—
88C8×Dic3—22no—
89C2×#121—31no—
90C2×#117—31no—
91#482—61no—
92#482—61no—
93#481—61no—
94#480—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98#428—31no—
99#428—31no—
100#424—61no—
101#418—31no—
102#418—31no—
103#407—31no—
104#407C2×D311yescharacteristic
105C2×C22⋊C8C2×D311yescharacteristic
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108#380—31no—
109#340—61no—
110#340—61no—
111#327—61no—
112#327—61no—
113#323—62no—
114C24×C3#4111yescharacteristic
115C23×D3—21no—
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×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—81no—
127C2×#67—81no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—41no—
135C2×D12—41no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—21no—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147#217—21no—
148#217—21no—
149#217—21no—
150#217—21no—
151#217—81no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3C22⋊C411yescharacteristic
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163C4⋊Dic3C22⋊C411yescharacteristic
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—22no—
171C22×D4—31no—
172C22×D4—61no—
173C23×C4C22×D311yescharacteristic
174C2×M16—61no—
175C2×M16C4×D311yescharacteristic
176C2×M16C4×D311yescharacteristic
177C22×C8—21no—
178C22×C8—21no—
179#121—61no—
180#121—61no—
181#121—61no—
182#121—61no—
183#117—31no—
184#117—31no—
185#117—31no—
186#117—31no—
187#117—61no—
188#117—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C42—61no—
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203#102—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—32no—
213C22⋊C8—61no—
214C22⋊C8D1211yescharacteristic
215C22⋊C8D1211yescharacteristic
216#97—62no—
217C4×C8—62no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12311yescharacteristic
226C22×D3—41no—
227C22×D3—42no—
228C22×D3—81no—
229C22×D3—81no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243#67—82no—
244#67—81no—
245#67—81no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258D12—41no—
259D12—41no—
260D12—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3×C8—41no—
264C3×C8—41no—
265C3×C8—81no—
266C3⋊C8—42no—
267C24—61no—
268C24#23911yescharacteristic
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—241no—
276C2×D4—241no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—121no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4C2×D1211yescharacteristic
285C22×C4#23511yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289M16—41no—
290M16—41no—
291M16—121no—
292M16—121no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—62no—
303C4⋊C4—31no—
304C4⋊C4—31no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—61no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—241no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#47111yescharacteristic
336C22×C3#47211yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—162no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4#41011yescharacteristic
345C3×C4#41011yescharacteristic
346Dic3—41no—
347Dic3—41no—
348Dic3—41no—
349Dic3—41no—
350Dic3—81no—
351Dic3—161no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—122no—
357C23—121no—
358C23—241no—
359C23—241no—
360C23#89311yescharacteristic
361C23#89411yescharacteristic
362C23#89511yescharacteristic
363D4—121no—
364D4—121no—
365D4—241no—
366D4—242no—
367D4—241no—
368D4—241no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4C4×D1211yescharacteristic
391C2×C4D3×C22⋊C411yescharacteristic
392C2×C4#88211yescharacteristic
393C2×C4#88411yescharacteristic
394C8—41no—
395C8—41no—
396C8—41no—
397C8—41no—
398C8—81no—
399C8—122no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3#188111yescharacteristic
405C2×C3#189911yescharacteristic
406C2×C3#190711yescharacteristic
407D3—161no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—241no—
419C22—241no—
420C22—241no—
421C22—482no—
422C22#466511yescharacteristic
423C22#489711yescharacteristic
424C22#489811yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—121no—
429C4—121no—
430C4—121no—
431C4—121no—
432C4—241no—
433C4—481no—
434C4#474011yescharacteristic
435C4#474011yescharacteristic
436C3#934811yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—481no—
442C2#6894711yescharacteristic
443C2#6894811yescharacteristic
444C2#6895411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1298C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1298D3
hypocenterC3#9348
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #78584C2semidirect
4. #69926C2semidirect
6. C3×#1298C2semidirect
7. #67684C2semidirect
16. #5131C4semidirect
17. #5131C4semidirect
18. #5121C4semidirect
19. #5121C4semidirect
40. #1298D3semidirect
436. C3#9348semidirect

Maximal subgroups

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

subgroupindex
#786242
#785842
#699442
#699262
#699212
C3×#12982
#676842
#93483