Finite Group Databasebeta

← #10692489

Subgroups — #10692489

Order 768.

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

normal subgroups only

This group has 530 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#79349C211yesmaximal, characteristic
2#74262C211yesmaximal, characteristic, complemented
3#69867C211yesmaximal, characteristic, complemented
4#69860C211yesmaximal, characteristic, complemented
5#69564C211yesmaximal, characteristic, complemented
6C3×#1288C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9381—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#383C2211yescharacteristic
14C2×#825—21no—
15C2×#825—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4926C2211yescharacteristic
18#4844—22no—
19#4738—21no—
20#4732—21no—
21#4714C412yescomplemented
22#4712C412yescomplemented
23#4670—21no—
24#4669—21no—
25#4668—21no—
26#4666—21no—
27#4665—21no—
28#4646—21no—
29#4464—22no—
30#4451—21no—
31#4451—21no—
32#3213—31no—
33#2337—31no—
34#1980—31no—
35#1527—31no—
36#1514—31no—
37#1288D311yescharacteristic, complemented
38#1285—31no—
39C2×#235—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×D24—22no—
58C2×#204—22no—
59#875—21no—
60#875—21no—
61#875—21no—
62#875C2×C412yes—
63#875D412yes—
64#874C2×C411yescharacteristic, complemented
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic, complemented
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97C2×C412yescomplemented
72C3×C4×C8—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95D411yescharacteristic
76C3×#95D411yescharacteristic
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#825—42no—
82#825—42no—
83#821—22no—
84#816—22no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#532—31no—
88C2×#124—31no—
89#496—32no—
90#489—32no—
91#463—62no—
92#448—61no—
93#447—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96C2×#104—61no—
97C2×#104—61no—
98C4×M16C2×D311yescharacteristic
99#391—61no—
100#388—61no—
101#388—61no—
102#386—61no—
103#385—61no—
104#383C2×D311yescharacteristic
105C4×C4⋊C4—31no—
106C4×C22⋊C4—61no—
107#329—62no—
108#327—61no—
109#327—61no—
110C23×D3—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120#235—41no—
121#235—41no—
122#235—41no—
123#235—41no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—81no—
129C2×D12C22⋊C412yes—
130C2×C4×D3—42no—
131C2×C4×D3—41no—
132C2×C4×D3—81no—
133C2×C4×D3—81no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—81no—
137C2×Dic6C22⋊C412yes—
138C3×M16—22no—
139C3×M16—41no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—42no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—81no—
154(C2×D3)⋊C4—81no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—42no—
158Dic3⋊C4—81no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3—81no—
162C2×C3⋊C8—41no—
163D24—82no—
164#204—82no—
165C2×#41—61no—
166C22×Q8—31no—
167C22×D4—31no—
168C23×C4—61no—
169C2×SD16—62no—
170C2×D8—62no—
171C2×M16—21no—
172C4⋊Q8—31no—
173#127—31no—
174#124—32no—
175#124—32no—
176#124—61no—
177#124—61no—
178#124—61no—
179#121—61no—
180#121—61no—
181C4×D4—61no—
182C4×D4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—61no—
196#104—122no—
197#104—122no—
198#103—62no—
199#102—62no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202#97—21no—
203#97—61no—
204#97C4×D312yescomplemented
205C4×C8—21no—
206#95—21no—
207#95—21no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95D1211yescharacteristic
212#95D1211yescharacteristic
213C23×C3#12011yescharacteristic
214C22×D3—22no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12311yescharacteristic
238#67—81no—
239#67—81no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250D12—42no—
251D12—42no—
252D12—41no—
253D12—41no—
254D12—81no—
255D12—81no—
256C4×D3—84no—
257C4×D3—81no—
258C4×D3—81no—
259Dic6—42no—
260Dic6—41no—
261Dic6—41no—
262Dic6—81no—
263C3×C8—42no—
264C3×C8—42no—
265C3⋊C8—81no—
266C24—61no—
267#41—121no—
268#41—121no—
269#41—121no—
270#41—121no—
271C2×Q8—32no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—241no—
275C2×D4—32no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—121no—
281C2×D4—241no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—122no—
288C22×C4—121no—
289C22×C4—241no—
290C22×C4—241no—
291C22×C4C2×D1211yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294SD16—242no—
295D8—242no—
296M16—22no—
297M16—41no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—121no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—241no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—122no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—241no—
313C22⋊C4—241no—
314C42—121no—
315C42—121no—
316C42—241no—
317C42C2×C4×D311yescharacteristic
318C42#23511yescharacteristic
319C42D3×D411yescharacteristic
320C42#23911yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3#38611yescharacteristic
324C22×C3#45311yescharacteristic
325C22×C3#45411yescharacteristic
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—42no—
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—81no—
343C3×C4—81no—
344Dic3—42no—
345Dic3—42no—
346Dic3—81no—
347Dic3—81no—
348C23—62no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23—121no—
356C23#88211yescharacteristic
357Q8—122no—
358Q8—121no—
359Q8—121no—
360Q8—241no—
361D4—122no—
362D4—122no—
363D4—121no—
364D4—121no—
365D4—241no—
366D4—241no—
367D4—241no—
368D4—241no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4—244no—
392C2×C4C4×D1211yescharacteristic
393C2×C4D3×C22⋊C411yescharacteristic
394C2×C4#88411yescharacteristic
395C2×C4#89311yescharacteristic
396C2×C4#89411yescharacteristic
397C2×C4#89511yescharacteristic
398C8—42no—
399C8—42no—
400C8—241no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3#188311yescharacteristic
404C2×C3#190611yescharacteristic
405C2×C3#191111yescharacteristic
406D3—82no—
407D3—161no—
408C22—21no—
409C22—21no—
410C22—122no—
411C22—122no—
412C22—122no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22#466511yescharacteristic
421C22#482211yescharacteristic
422C22#482711yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—81no—
430C4—81no—
431C4—122no—
432C4—122no—
433C4—241no—
434C4—241no—
435C3#938111yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—242no—
439C2—481no—
440C2#6870511yescharacteristic
441C2#6877611yescharacteristic
442C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radical#79349C2
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9381
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
2. #74262C2semidirect
3. #69867C2semidirect
4. #69860C2semidirect
5. #69564C2semidirect
6. C3×#1288C2semidirect
7. #67749C2semidirect
21. #4714C4semidirect
22. #4712C4semidirect
37. #1288D3semidirect
64. #874C2×C4semidirect
67. #869C2×C4semidirect
71. C3×#97C2×C4semidirect
204. #97C4×D3semidirect
435. C3#9381semidirect

Maximal subgroups

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

subgroupindex
#793492
#742622
#698672
#698602
#695642
C3×#12882
#677492
#93813