Finite Group Databasebeta

← #10693065

Subgroups — #10693065

Order 768.

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

normal subgroups only

This group has 544 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#78514C211yesmaximal, characteristic
2#74308C211yesmaximal, characteristic
3#70826C211yesmaximal, characteristic
4#70802C211yesmaximal, characteristic, complemented
5#69857C211yesmaximal, characteristic, complemented
6C3×#1288C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9491—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#872C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×#383C2211yescharacteristic
15Dic3×M16—21no—
16#4926C2211yescharacteristic
17C4×#208C2211yescharacteristic
18#4708—22no—
19#4689—22no—
20#4674C411yescharacteristic
21#4673C411yescharacteristic
22#4670—21no—
23#4668—21no—
24#4661C411yescharacteristic
25#4656C411yescharacteristic
26#4653—21no—
27#4653—21no—
28#4484—22no—
29#4469—21no—
30#4468—21no—
31#4456—22no—
32#2873—31no—
33#2599—31no—
34#1491—31no—
35#1491—31no—
36#1464—31no—
37#1288—31no—
38#1288D311yescharacteristic, complemented
39C22×C4×D3D411yescharacteristic
40C22×Dic6C2×C411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4C2×C411yescharacteristic
46C2×(C2×D3)⋊C4D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4C2×C411yescharacteristic
52C2×Dic3⋊C4D411yescharacteristic
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56D3×M16—24no—
57D3×M16—24no—
58C2×ΓL(1,𝔽25)—22no—
59C2×D3×C8—22no—
60#872—21no—
61#872—21no—
62#872—21no—
63#872—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97—21no—
72C3×C4×C8D412yes—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95C2×C411yescharacteristic
76C3×#95C2×C411yescharacteristic
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#815—22no—
83C8×Dic3—22no—
84C4⋊(C3⋊C8)—21no—
85#803—21no—
86#803—21no—
87C4×C3⋊C8D412yes—
88C22×M16—61no—
89C2×C4×Q8—31no—
90C2×#117—31no—
91#432—31no—
92#432C2×D311yescharacteristic
93#405—62no—
94#405—62no—
95C4×M16—31no—
96C4×M16—61no—
97C4×M16C2×D311yescharacteristic
98#398—31no—
99#397—31no—
100#395—31no—
101#393—31no—
102#388—61no—
103#387—61no—
104#387—61no—
105#385—61no—
106#383C2×D311yescharacteristic
107Q8⋊C8—62no—
108Q8⋊C8—62no—
109#324—61no—
110#323—61no—
111C23×D3C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C22×Dic3C22⋊C411yescharacteristic
122C2×C4×D3—22no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—22no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—42no—
129C3×M16—22no—
130C3×M16—22no—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—42no—
142(C2×D3)⋊C4—42no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—42no—
146C4⋊Dic3—42no—
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157#208—22no—
158#208—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161ΓL(1,𝔽25)—28no—
162D3×C8—28no—
163C22×Q8—31no—
164C23×C4—31no—
165C2×M16—21no—
166C2×M16—62no—
167C2×M16—64no—
168C2×M16—64no—
169C2×M16—61no—
170C22×C8—62no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175#117—61no—
176#117—61no—
177#117—61no—
178#117—61no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C42—31no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192C4⋊C8—21no—
193C4⋊C8—61no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196#97—21no—
197#97—21no—
198#97—62no—
199#97—61no—
200#97—61no—
201C4×C8—32no—
202C4×C8—62no—
203C4×C8D1212yes—
204#95—21no—
205#95—21no—
206#95—61no—
207#95—61no—
208#95—61no—
209#95—61no—
210#95C4×D311yescharacteristic
211#95C4×D311yescharacteristic
212C23×C3C2×C22⋊C411yescharacteristic
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3#10412yes—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—22no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3#10412yes—
247C4×D3—22no—
248C4×D3—22no—
249C4×D3—24no—
250Dic6—22no—
251Dic6—22no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—24no—
255C3×C8—81no—
256C3⋊C8—24no—
257C3⋊C8—81no—
258C24—31no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—122no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4C2×C4×D311yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276M16—22no—
277M16—22no—
278M16—68no—
279M16—62no—
280M16—62no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—68no—
284C2×C8—62no—
285C2×C8—121no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—122no—
293C4⋊C4—122no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—122no—
301C22⋊C4—122no—
302C42—61no—
303C42—61no—
304C42—121no—
305C42—121no—
306C42C2×D1211yescharacteristic
307C42#23511yescharacteristic
308C42D3×D411yescharacteristic
309C42#23911yescharacteristic
310C22×C3—21no—
311C22×C3—21no—
312C22×C3#38611yescharacteristic
313C22×C3C2×#10411yescharacteristic
314C22×C3C2×#10411yescharacteristic
315C2×D3—22no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—42no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—81no—
328Dic3—22no—
329Dic3—42no—
330Dic3—41no—
331Dic3—41no—
332Dic3—81no—
333C23—32no—
334C23—61no—
335C23—61no—
336C23—121no—
337C23—121no—
338C23D3×C22⋊C411yescharacteristic
339Q8—62no—
340Q8—62no—
341Q8—121no—
342Q8—121no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—32no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—64no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—62no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4C4×D1211yescharacteristic
370C2×C4#88211yescharacteristic
371C2×C4#88411yescharacteristic
372C2×C4#89311yescharacteristic
373C2×C4#89411yescharacteristic
374C2×C4#89511yescharacteristic
375C8—24no—
376C8—64no—
377C8—81no—
378C8—241no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3#189111yescharacteristic
382C2×C3#189611yescharacteristic
383C2×C3#190911yescharacteristic
384D3—42no—
385C22—21no—
386C22—21no—
387C22—62no—
388C22—122no—
389C22—121no—
390C22—121no—
391C22—241no—
392C22#466511yescharacteristic
393C22D3×#10411yescharacteristic
394C22D3×#10411yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—62no—
402C4—81no—
403C4—81no—
404C4—121no—
405C4—122no—
406C4—121no—
407C4—241no—
408C3#949111yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—122no—
412C2#6871211yescharacteristic
413C2#6875611yescharacteristic
414C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9491
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
4. #70802C2semidirect
5. #69857C2semidirect
6. C3×#1288C2semidirect
7. #67633C2semidirect
38. #1288D3semidirect
408. C3#9491semidirect

Maximal subgroups

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

subgroupindex
#785142
#743082
#708262
#708022
#698572
C3×#12882
#676332
#94913