Finite Group Databasebeta

← #10692815

Subgroups — #10692815

Order 768.

576 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 89 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#78531C211yesmaximal, characteristic
2#73395C211yesmaximal, characteristic, complemented
3#70832C211yesmaximal, characteristic, complemented
4#70808C211yesmaximal, characteristic, complemented
5#69432C211yesmaximal, characteristic, complemented
6C3×#1300C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9441—31nomaximal, Sylow
9C2×#909—21no—
10C2×#908—21no—
11#5477C2211yescharacteristic
12C2×#875C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C3×#376C2211yescharacteristic
16#4940—21no—
17C4×(C2×D3)⋊C4—21no—
18#4932C2211yescharacteristic
19#4926C2211yescharacteristic
20#4715C412yescomplemented
21#4712C412yescomplemented
22#4696—22no—
23#4669—21no—
24#4666—21no—
25#4665—21no—
26#4646—21no—
27#4470—21no—
28#4470—21no—
29#4464—22no—
30#4451—21no—
31#4451—21no—
32#3153—31no—
33#2372—31no—
34#1533—31no—
35#1505—31no—
36#1505—31no—
37#1300—31no—
38#1300D311yescharacteristic, complemented
39C2×#235—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56#909—42no—
57#909—42no—
58#908—42no—
59#908—42no—
60C2×Dic12—22no—
61C2×D24—22no—
62C2×#204—22no—
63C2×#204—22no—
64#875—21no—
65#875—21no—
66#875—21no—
67#875C2×C412yes—
68#875D412yes—
69#874C2×C411yescharacteristic, complemented
70C4×D12—21no—
71C4×D12—21no—
72#869C2×C411yescharacteristic, complemented
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#97C2×C412yescomplemented
77C3×C4×C8—21no—
78C3×#95—21no—
79C3×#95D411yescharacteristic
80C3×#95D411yescharacteristic
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95D411yescharacteristic
85C3⋊#95D411yescharacteristic
86#830—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89C2×#137—61no—
90C2×AGL(1,ℤ/8)—61no—
91#532—31no—
92C2×#124—31no—
93#497—32no—
94#496—32no—
95#495—62no—
96#432—31no—
97#432C2×D311yescharacteristic
98C4×M16C2×D311yescharacteristic
99#391—61no—
100#388—61no—
101#388—61no—
102#386—61no—
103C4×C22⋊C4—61no—
104C4×C22⋊C4—61no—
105#376—31no—
106#376C2×D311yescharacteristic
107#329—62no—
108#327—61no—
109#327—61no—
110#327—61no—
111#327—61no—
112C23×D3—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—41no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—41no—
123#235—41no—
124#235—41no—
125#235—41no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12C22⋊C412yes—
133C2×C4×D3—42no—
134C2×C4×D3—42no—
135C2×C4×D3—41no—
136C2×C4×D3—81no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—41no—
140C2×Dic6—41no—
141C2×Dic6C22⋊C412yes—
142C3×M16—22no—
143C3×M16—22no—
144C2×C3×C8—22no—
145C2×C3×C8—41no—
146C3×C42—42no—
147C3×C42C22×C411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150C3×C42#4111yescharacteristic
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—22no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—22no—
155(C2×D3)⋊C4—42no—
156(C2×D3)⋊C4—42no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—81no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—81no—
163C4×Dic3—42no—
164C4×Dic3—81no—
165C2×C3⋊C8—41no—
166Dic12—44no—
167D24—44no—
168#204—44no—
169#204—44no—
170C2×#41—61no—
171C22×Q8—31no—
172C22×D4—31no—
173C23×C4—61no—
174#137—122no—
175#137—122no—
176AGL(1,ℤ/8)—122no—
177AGL(1,ℤ/8)—122no—
178C2×Q16—62no—
179C2×SD16—62no—
180C2×SD16—62no—
181C2×D8—62no—
182C2×M16—21no—
183C4⋊Q8—31no—
184#127—31no—
185#124—32no—
186#124—32no—
187#124—61no—
188#124—61no—
189#124—61no—
190#121—61no—
191#121—61no—
192C4×D4—61no—
193C4×D4—61no—
194C2×C4⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C42—21no—
201C2×C42—61no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—61no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208#97—21no—
209#97—61no—
210#97C4×D312yescomplemented
211C4×C8—21no—
212#95—21no—
213#95—31no—
214#95—31no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95D1211yescharacteristic
219#95D1211yescharacteristic
220C23×C3#12011yescharacteristic
221C22×D3—22no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12311yescharacteristic
245#67—81no—
246#67—81no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257D12—42no—
258D12—41no—
259D12—41no—
260D12—81no—
261D12—81no—
262D12—81no—
263C4×D3—84no—
264C4×D3—81no—
265C4×D3—81no—
266Dic6—42no—
267Dic6—41no—
268Dic6—41no—
269Dic6—81no—
270C3×C8—24no—
271C3×C8—81no—
272C3⋊C8—81no—
273C24—61no—
274#41—121no—
275#41—121no—
276#41—121no—
277#41—121no—
278C2×Q8—32no—
279C2×Q8—61no—
280C2×Q8—61no—
281C2×Q8—121no—
282C2×Q8—121no—
283C2×D4—32no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—122no—
297C22×C4—122no—
298C22×C4—121no—
299C22×C4—241no—
300C22×C4C2×D1211yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4D3×D411yescharacteristic
303Q16—124no—
304SD16—124no—
305SD16—124no—
306D8—124no—
307M16—22no—
308M16—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—121no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—241no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—122no—
320C22⋊C4—122no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C22⋊C4—241no—
324C42—42no—
325C42—122no—
326C42—241no—
327C42C2×C4×D311yescharacteristic
328C42#23511yescharacteristic
329C42D3×D411yescharacteristic
330C42#23911yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3#38611yescharacteristic
334C22×C3#45311yescharacteristic
335C22×C3#45311yescharacteristic
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—42no—
353C3×C4—81no—
354Dic3—42no—
355Dic3—42no—
356Dic3—81no—
357Dic3—81no—
358C23—62no—
359C23—121no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23#88211yescharacteristic
367Q8—122no—
368Q8—121no—
369Q8—121no—
370Q8—241no—
371D4—122no—
372D4—121no—
373D4—121no—
374D4—241no—
375D4—241no—
376D4—241no—
377D4—241no—
378D4—241no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4—244no—
402C2×C4C4×D1211yescharacteristic
403C2×C4D3×C22⋊C411yescharacteristic
404C2×C4#88411yescharacteristic
405C2×C4#89311yescharacteristic
406C2×C4#89411yescharacteristic
407C2×C4#89511yescharacteristic
408C8—24no—
409C8—81no—
410C8—241no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3#188711yescharacteristic
414C2×C3#190611yescharacteristic
415C2×C3#190611yescharacteristic
416D3—82no—
417D3—161no—
418C22—21no—
419C22—21no—
420C22—122no—
421C22—122no—
422C22—122no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22—241no—
429C22—241no—
430C22#466511yescharacteristic
431C22#482211yescharacteristic
432C22#482211yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—81no—
441C4—122no—
442C4—122no—
443C4—241no—
444C4—241no—
445C3#944111yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—242no—
449C2—481no—
450C2#6876011yescharacteristic
451C2#6877811yescharacteristic
452C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radical#78531C2
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9441
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
2. #73395C2semidirect
3. #70832C2semidirect
4. #70808C2semidirect
5. #69432C2semidirect
6. C3×#1300C2semidirect
7. #67665C2semidirect
20. #4715C4semidirect
21. #4712C4semidirect
38. #1300D3semidirect
69. #874C2×C4semidirect
72. #869C2×C4semidirect
76. C3×#97C2×C4semidirect
210. #97C4×D3semidirect
445. C3#9441semidirect

Maximal subgroups

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

subgroupindex
#785312
#733952
#708322
#708082
#694322
C3×#13002
#676652
#94413