Finite Group Databasebeta

← #10695983

Subgroups — #10695983

Order 768.

566 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 85 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#79343C211yesmaximal, characteristic
2#79318C211yesmaximal, characteristic, complemented
3#73806C211yesmaximal, characteristic, complemented
4#70812C211yesmaximal, characteristic, complemented
5#70799C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic, complemented
7C3×#1285C211yesmaximal, characteristic, complemented
8#10005—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#4987—21no—
16#4987—21no—
17#4940—22no—
18C2×#805—22no—
19C4×#208C2211yescharacteristic
20#4847—22no—
21#4843—21no—
22#4842—21no—
23#4736—21no—
24#4735—21no—
25#4735—21no—
26#4732—21no—
27#4713—21no—
28#4711—22no—
29#4702—21no—
30#4672—21no—
31#4659—21no—
32#4483—22no—
33#4465—22no—
34#3234—31no—
35#3109—31no—
36#2562—31no—
37#1980—31no—
38#1528—31no—
39#1490—31no—
40#1285D311yescharacteristic, complemented
41C2×#235—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42D412yes—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×#208—21no—
58C2×D24—22no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#869D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×#97—21no—
77C3×#97D412yes—
78C3×#95—21no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81#821—22no—
82#821—22no—
83#821—22no—
84#821—22no—
85#818—22no—
86#817—22no—
87#817—22no—
88#805—44no—
89#803—21no—
90C4×C3⋊C8—21no—
91#532—31no—
92C2×#125—31no—
93C2×C4×D4—31no—
94#501—61no—
95#490—61no—
96#481—61no—
97#481—61no—
98#480—61no—
99#478—62no—
100#476—61no—
101#459—61no—
102#447—61no—
103#442—62no—
104#432C2×D311yescharacteristic
105#432C2×D311yescharacteristic
106C2×#104—62no—
107C4×M16—31no—
108#400—61no—
109#394—61no—
110#388—62no—
111#380—61no—
112#380—61no—
113C4×C4⋊C4C2×D311yescharacteristic
114#330—62no—
115D4⋊C8—62no—
116C23×D3—21no—
117C22×C3×C4—22no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125#235—42no—
126#235—42no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—42no—
133C2×C4×D3—42no—
134C2×C4×D3—42no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C3×C4⋊C4—42no—
142C3×C42—42no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147(C2×D3)⋊C4—44no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—82no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165#208—22no—
166#208—41no—
167C2×C3⋊C8—41no—
168D24—44no—
169C2×#41—61no—
170C22×D4—31no—
171C23×C4—61no—
172C2×D8—62no—
173C2×M16—61no—
174C4⋊Q8—31no—
175#127—31no—
176#125—61no—
177#125—61no—
178#125—61no—
179#125—61no—
180#124—61no—
181#121—61no—
182#121—61no—
183C4×D4—61no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—62no—
198C2×C42D1212yes—
199C2×C42C22×D311yescharacteristic
200#107—62no—
201#106—62no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8D1212yes—
205#104—124no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212#97—21no—
213#97—61no—
214#97D1212yes—
215C4×C8—61no—
216#95—21no—
217#95—62no—
218#95—62no—
219C23×C3#12011yescharacteristic
220C22×D3—22no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241#67—81no—
242#67—81no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252D12—22no—
253D12—22no—
254D12—41no—
255D12—41no—
256D12—42no—
257D12—42no—
258D12—42no—
259C4×D3—82no—
260C4×D3—82no—
261C4×D3—82no—
262Dic6—42no—
263C3×C8—42no—
264C3×C8—42no—
265C3⋊C8—42no—
266C24—61no—
267#41—122no—
268#41—122no—
269C2×Q8—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—122no—
276C2×D4—121no—
277C22×C4—22no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—122no—
282C22×C4—122no—
283C22×C4—121no—
284C22×C4—121no—
285C22×C4—121no—
286C22×C4C2×D1211yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289D8—124no—
290M16—62no—
291M16—121no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—121no—
295C4⋊C4—42no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—124no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—242no—
314C42—42no—
315C42C2×D1211yescharacteristic
316C42C2×D1211yescharacteristic
317C42D3×D411yescharacteristic
318C42#23911yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3#39211yescharacteristic
322C22×C3#45311yescharacteristic
323C22×C3#45411yescharacteristic
324C2×D3—42no—
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—81no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4#45412yes—
341Dic3—81no—
342Dic3—81no—
343Dic3—81no—
344Dic3—81no—
345C23—62no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23#88211yescharacteristic
354Q8—122no—
355D4—62no—
356D4—62no—
357D4—121no—
358D4—121no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—241no—
363D4—241no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—242no—
381C2×C4—242no—
382C2×C4—242no—
383C2×C4#87411yescharacteristic
384C2×C4#88211yescharacteristic
385C2×C4#88211yescharacteristic
386C2×C4#89511yescharacteristic
387C2×C4#89511yescharacteristic
388C2×C4#89511yescharacteristic
389C8—42no—
390C8—42no—
391C8—122no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3#200311yescharacteristic
395C2×C3#200611yescharacteristic
396C2×C3#201011yescharacteristic
397D3—82no—
398D3—161no—
399C22—21no—
400C22—21no—
401C22—122no—
402C22—122no—
403C22—122no—
404C22—241no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22#467111yescharacteristic
412C22#474711yescharacteristic
413C22#475311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—82no—
420C4—241no—
421C4—241no—
422C4—241no—
423C4—241no—
424C4#482712yes—
425C3#1000511yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—242no—
429C2—481no—
430C2#6870311yescharacteristic
431C2#6876711yescharacteristic
432C2#6900811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#392
prime radical#79343C2
hypercenter#1285D3
hypocenterC3#10005
LCSG ▷ C2×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. #79318C2semidirect
3. #73806C2semidirect
4. #70812C2semidirect
5. #70799C2semidirect
6. #69492C2semidirect
7. C3×#1285C2semidirect
40. #1285D3semidirect
425. C3#10005semidirect

Maximal subgroups

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

subgroupindex
#793432
#793182
#738062
#708122
#707992
#694922
C3×#12852
#100053