Finite Group Databasebeta

← #10691018

Subgroups — #10691018

Order 768.

458 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 95 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#79355C211yesmaximal, characteristic
2#79315C211yesmaximal, characteristic, complemented
3#70806C211yesmaximal, characteristic
4#69475C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#9164—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#810—21no—
15#4925C2211yescharacteristic
16C2×#803C2211yescharacteristic
17#4850C411yescharacteristic, complemented
18#4850C411yescharacteristic, complemented
19#4848C411yescharacteristic, complemented
20#4848C411yescharacteristic, complemented
21#4831—21no—
22#4830—21no—
23#4738—21no—
24#4736—21no—
25#4719—21no—
26#4718—21no—
27#4710—21no—
28#4710—21no—
29#4701—22no—
30#4529—21no—
31#4464—22no—
32#4451—21no—
33#4451—21no—
34#3112—31no—
35#2984—31no—
36#1763—31no—
37#1530—31no—
38#1502—31no—
39#1302—31no—
40#1285D311yescharacteristic, complemented
41C2×#235—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52C2×#204—21no—
53C2×#204—21no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869C2×C411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6C2×C411yescharacteristic
63C4×Dic6D411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8C2×C411yescharacteristic
67C3×C4⋊C8C2×C411yescharacteristic
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×#95—21no—
73#830D411yescharacteristic
74#830D411yescharacteristic
75#821—21no—
76#821—21no—
77#818—21no—
78#818—21no—
79#816—21no—
80#816—21no—
81#815—22no—
82Dic3⋊C8—22no—
83Dic3⋊C8—22no—
84C8×Dic3—22no—
85#810—24no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90#532—31no—
91C2×C4×Q8—31no—
92#480—61no—
93#477—31no—
94#477—31no—
95#460—31no—
96#460—31no—
97#448—61no—
98#446—61no—
99#446—62no—
100C8×Q8—61no—
101#440—61no—
102#440—61no—
103#432—61no—
104#432C2×D311yescharacteristic
105#432C2×D311yescharacteristic
106#431—31no—
107#431—61no—
108C2×#103—61no—
109C2×#97—31no—
110C4×C4⋊C4C2×D311yescharacteristic
111#340—61no—
112#329—62no—
113#327—61no—
114#327—61no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—41no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—81no—
123#235—81no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—22no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—41no—
132C2×Dic6C22⋊C411yescharacteristic
133C2×Dic6C22⋊C411yescharacteristic
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—41no—
139C3×C42—81no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3C22⋊C411yescharacteristic
150C4⋊Dic3C22⋊C411yescharacteristic
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164#204—42no—
165#204—42no—
166C2×#41—61no—
167C22×Q8—31no—
168C2×SD16—61no—
169C2×SD16—61no—
170C22×C8—61no—
171C4⋊Q8—31no—
172#127—31no—
173#124—61no—
174#121—61no—
175#121—61no—
176C4×Q8—31no—
177C4×Q8—31no—
178C4×Q8—31no—
179C4×Q8—31no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—61no—
191#107—61no—
192C4⋊C8—21no—
193C4⋊C8—62no—
194C4⋊C8—62no—
195C4⋊C8—61no—
196C4⋊C8C4×D311yescharacteristic
197C4⋊C8C4×D311yescharacteristic
198#103—64no—
199#103—61no—
200#103—61no—
201#102—61no—
202#102—61no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—31no—
206C22⋊C8—31no—
207#97—21no—
208#97—62no—
209#97—61no—
210#97—61no—
211#97D1212yes—
212C4×C8—62no—
213C4×C8—61no—
214#95—21no—
215C23×C3#12111yescharacteristic
216C22×D3—41no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12311yescharacteristic
231#67—81no—
232#67—81no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241D12—41no—
242D12—41no—
243D12—81no—
244D12—81no—
245C4×D3—82no—
246Dic6—21no—
247Dic6—21no—
248Dic6—21no—
249Dic6—42no—
250Dic6—81no—
251Dic6#10412yes—
252C3×C8—42no—
253C3×C8—41no—
254C3×C8—41no—
255C3⋊C8—41no—
256C3⋊C8—41no—
257C3⋊C8—42no—
258#41—241no—
259#41—241no—
260C2×Q8—31no—
261C2×Q8—31no—
262C2×Q8—62no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—121no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—121no—
270C22×C4—41no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4#23511yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277SD16—122no—
278SD16—122no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—62no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—22no—
288C4⋊C4—31no—
289C4⋊C4—31no—
290C4⋊C4—41no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—61no—
301C42—61no—
302C42—61no—
303C42—61no—
304C42—81no—
305C42C2×C4×D311yescharacteristic
306C42C2×D1211yescharacteristic
307C42D3×D411yescharacteristic
308C42#23911yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3#38611yescharacteristic
312C22×C3#47111yescharacteristic
313C22×C3#47211yescharacteristic
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—82no—
323C3×C4C2×#10411yescharacteristic
324C3×C4#42111yescharacteristic
325Dic3—22no—
326Dic3—41no—
327Dic3—41no—
328Dic3—41no—
329Dic3—81no—
330C23—121no—
331C23#88411yescharacteristic
332Q8—32no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—122no—
337Q8—241no—
338D4—121no—
339D4—121no—
340D4—241no—
341D4—241no—
342D4—241no—
343D4—241no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—122no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—242no—
361C2×C4C4×D1211yescharacteristic
362C2×C4D3×C22⋊C411yescharacteristic
363C2×C4#88211yescharacteristic
364C2×C4#89311yescharacteristic
365C2×C4#89411yescharacteristic
366C2×C4#89511yescharacteristic
367C8—42no—
368C8—41no—
369C8—41no—
370C8—122no—
371C8—121no—
372C8—121no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#187111yescharacteristic
376C2×C3#190511yescharacteristic
377C2×C3#190711yescharacteristic
378D3—161no—
379C22—21no—
380C22—21no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#466511yescharacteristic
385C22#474911yescharacteristic
386C22#475411yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—62no—
393C4—82no—
394C4—121no—
395C4—121no—
396C4—121no—
397C4—241no—
398C4D3×#10411yescharacteristic
399C4#482111yescharacteristic
400C3#916411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—481no—
404C2#6870711yescharacteristic
405C2#6875511yescharacteristic
406C2#6900611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9164
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
2. #79315C2semidirect
4. #69475C2semidirect
6. C3×#1285C2semidirect
7. #67751C2semidirect
17. #4850C4semidirect
18. #4850C4semidirect
19. #4848C4semidirect
20. #4848C4semidirect
40. #1285D3semidirect
400. C3#9164semidirect

Maximal subgroups

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

subgroupindex
#793552
#793152
#708062
#694752
#694692
C3×#12852
#677512
#91643