Finite Group Databasebeta

← #10692519

Subgroups — #10692519

Order 768.

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

normal subgroups only

This group has 582 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#83000C211yesmaximal, characteristic
2#78514C211yesmaximal, characteristic, complemented
3C3×#1527C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5#70826C211yesmaximal, characteristic, complemented
6#67748C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9387—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×D3×M16—21no—
11C2×#872C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5231C411yescharacteristic
15#5230C411yescharacteristic
16#5223C411yescharacteristic
17#5222C411yescharacteristic
18Dic3×M16—21no—
19#4957—21no—
20#4956—21no—
21#4945—21no—
22#4944—21no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4708—22no—
27#4689—22no—
28C3×#329—22no—
29#4540—22no—
30#4469—21no—
31#4468—21no—
32#2873—31no—
33#2610—31no—
34#1527—31no—
35#1527D311yescharacteristic, complemented
36#1464—31no—
37#1300—31no—
38#1300—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×#67—21no—
42C22×C4×D3D411yescharacteristic
43C2×C3×M16—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4C2×C411yescharacteristic
56C2×(C2×D3)⋊C4D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4C2×C411yescharacteristic
59C2×Dic3⋊C4D411yescharacteristic
60C2×C4×Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C2×#208—21no—
63D3×M16—24no—
64D3×M16—24no—
65C2×ΓL(1,𝔽25)—22no—
66C2×D3×C8—22no—
67#872—21no—
68#872—21no—
69#872—21no—
70#872—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yescomplemented
75C3×C4×C8—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95C2×C411yescharacteristic
80C3⋊#95C2×C411yescharacteristic
81#830—21no—
82#815—22no—
83C8×Dic3—22no—
84C4⋊(C3⋊C8)—21no—
85#803—21no—
86#803D412yescomplemented
87C4×C3⋊C8—21no—
88C22×M16—61no—
89C2×#124C2×D311yescharacteristic
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#395—31no—
99#394—31no—
100#393—31no—
101#392—31no—
102#390—61no—
103#388—61no—
104#388—61no—
105#386—61no—
106#376—31no—
107#329—22no—
108#329—62no—
109#324—61no—
110#323—61no—
111C24×C3—21no—
112C23×D3C22⋊C411yescharacteristic
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—81no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—81no—
119C22×C3×C4—21no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—42no—
124C2×#67—42no—
125C2×#67—81no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3C22⋊C411yescharacteristic
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C2×C4×D3—22no—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—22no—
135C3×M16—22no—
136C3×M16—22no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147#217—42no—
148#217—42no—
149#217—81no—
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154(C2×D3)⋊C4—81no—
155Dic3⋊C4—21no—
156Dic3⋊C4—21no—
157Dic3⋊C4—21no—
158Dic3⋊C4—21no—
159Dic3⋊C4—81no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3—42no—
163#208—22no—
164#208—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167ΓL(1,𝔽25)—28no—
168D3×C8—28no—
169C22×Q8C4×D311yescharacteristic
170C22×D4—61no—
171C22×D4C4×D311yescharacteristic
172C23×C4—31no—
173C2×M16—21no—
174C2×M16—62no—
175C2×M16—64no—
176C2×M16—64no—
177C2×M16—61no—
178C22×C8—62no—
179#124—21no—
180#124—21no—
181#124—21no—
182#124—21no—
183#117—61no—
184#117—61no—
185#117—61no—
186#117—61no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—31no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C42—31no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—61no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205#97—21no—
206#97—32no—
207#97—62no—
208#97—61no—
209#97#6712yescomplemented
210C4×C8—21no—
211C4×C8—62no—
212C4×C8—61no—
213#95—31no—
214#95—31no—
215#95—61no—
216#95—61no—
217#95—61no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3C2×C22⋊C411yescharacteristic
226C22×D3—21no—
227C22×D3—21no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3#10412yes—
231C3×Q8—81no—
232C3×Q8—81no—
233C3×D4—81no—
234C3×D4—81no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248#67—84no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—22no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3#10412yes—
262C4×D3—22no—
263C4×D3—22no—
264C4×D3—24no—
265C3×C8—24no—
266C3×C8—81no—
267C3⋊C8—24no—
268C3⋊C8—81no—
269C24—21no—
270C24—31no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—81no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—81no—
277C2×D4—122no—
278C2×D4—122no—
279C2×D4—241no—
280C22×C4—21no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4C2×C4×D311yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293M16—22no—
294M16—22no—
295M16—68no—
296M16—62no—
297M16—62no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—68no—
301C2×C8—62no—
302C2×C8—121no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—241no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—122no—
317C22⋊C4—122no—
318C22⋊C4—241no—
319C22⋊C4—241no—
320C42—61no—
321C42—61no—
322C42—122no—
323C42#23511yescharacteristic
324C42D3×D411yescharacteristic
325C42#23711yescharacteristic
326C42C2×#6711yescharacteristic
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#38611yescharacteristic
337C22×C3C2×#10411yescharacteristic
338C22×C3C2×#10411yescharacteristic
339C2×D3—22no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—42no—
343C2×D3—81no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—81no—
351Dic3—22no—
352Dic3—42no—
353Dic3—41no—
354Dic3—41no—
355Dic3—81no—
356C23—32no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—61no—
365C23—61no—
366C23—121no—
367C23—121no—
368C23D3×C22⋊C411yescharacteristic
369Q8—81no—
370Q8—81no—
371D4—81no—
372D4—81no—
373D4—244no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—32no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—64no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—62no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4#67⋊C411yescharacteristic
398C2×C4#88311yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4C4×#6711yescharacteristic
401C2×C4#93711yescharacteristic
402C2×C4#93811yescharacteristic
403C8—24no—
404C8—64no—
405C8—81no—
406C8—241no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—161no—
410C2×C3#188411yescharacteristic
411C2×C3#189611yescharacteristic
412C2×C3#191411yescharacteristic
413D3—42no—
414C22—21no—
415C22—21no—
416C22—62no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—122no—
425C22—121no—
426C22—121no—
427C22—241no—
428C22D3×#10411yescharacteristic
429C22D3×#10411yescharacteristic
430C22#495611yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—62no—
438C4—81no—
439C4—122no—
440C4—121no—
441C4—121no—
442C4—241no—
443C3#938711yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—122no—
447C2—161no—
448C2#7008911yescharacteristic
449C2#7038211yescharacteristic
450C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#386
prime radical#5223C4
cosocleC2×C3×C42C23
hypercenter#1527D3
hypocenterC3#9387
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #78514C2semidirect
3. C3×#1527C2semidirect
4. #71718C2semidirect
5. #70826C2semidirect
6. #67748C2semidirect
7. #67665C2semidirect
35. #1527D3semidirect
74. C3×#97D4semidirect
86. #803D4semidirect
209. #97#67semidirect
443. C3#9387semidirect

Maximal subgroups

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

subgroupindex
#830002
#785142
C3×#15272
#717182
#708262
#677482
#676652
#93873