Finite Group Databasebeta

← #10695846

Subgroups — #10695846

Order 768.

544 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 81 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#78625C211yesmaximal, characteristic, complemented
2#78624C211yesmaximal, characteristic
3#74145C211yesmaximal, characteristic
4#69981C211yesmaximal, characteristic, complemented
5#69978C211yesmaximal, characteristic
6#69534C211yesmaximal, characteristic, complemented
7C3×#1306C211yesmaximal, characteristic, complemented
8#9984—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#407C2211yescharacteristic
14C3×#381C2211yescharacteristic
15#5211C2211yescharacteristic
16#5131—21no—
17#5121—21no—
18#5121—21no—
19#5120—21no—
20#4961—22no—
21#4737—22no—
22#4736—21no—
23#4735—21no—
24#4723—22no—
25#4722—21no—
26#4721—21no—
27#4677—21no—
28#4673—21no—
29#4672—21no—
30#4661—21no—
31#4527—22no—
32#4470—22no—
33#4467—22no—
34#3095—31no—
35#3094—31no—
36#2816—31no—
37#1820—31no—
38#1511—31no—
39#1510—31no—
40#1306D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×M16—21no—
47C2×C3×C22⋊C4D412yes—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58#926—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4⋊Dic3D411yescharacteristic
67#924—21no—
68#924—21no—
69#924—21no—
70#924—21no—
71C2×Dic3⋊C4—21no—
72C2×C4×Dic3—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×C22⋊C8D412yes—
77C3×#95—21no—
78C3×#95—21no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#821—22no—
83#821—22no—
84#818—22no—
85#818—22no—
86#817—22no—
87#817—22no—
88#816—22no—
89#816—22no—
90C2×C22⋊Q8—31no—
91C2×#121—31no—
92C2×#117—31no—
93#485—62no—
94#484—61no—
95#483—61no—
96#482—62no—
97#481—61no—
98#480—61no—
99#428—61no—
100#428—61no—
101#419—61no—
102#418—61no—
103#407C2×D311yescharacteristic
104#407C2×D311yescharacteristic
105#399—61no—
106#398—61no—
107#394—61no—
108#394—62no—
109#393—61no—
110#381—31no—
111#381C2×D311yescharacteristic
112#342—62no—
113#328—62no—
114#327—62no—
115C24×C3C2×D411yescharacteristic
116C23×D3—21no—
117C22×C3×C4—22no—
118C22×C3×C4—42no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—82no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—41no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×M16—42no—
141C3×M16—42no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C22⋊C4—42no—
145#217—41no—
146#217—41no—
147#217—41no—
148#217—41no—
149#217—42no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—82no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—82no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166C4×Dic3—41no—
167C22×Q8—31no—
168C22×D4—31no—
169C22×D4—61no—
170C23×C4C22×D311yescharacteristic
171C2×M16—21no—
172C2×M16—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180#121—61no—
181#117—61no—
182#117—61no—
183#117—61no—
184#117—61no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—62no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4D1212yes—
198C2×C42—61no—
199#107—62no—
200#107—62no—
201#106—62no—
202#106—62no—
203#103—62no—
204#103—62no—
205#102—62no—
206#102—62no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8D1212yes—
210C22⋊C8D1212yes—
211#95—21no—
212#95—21no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3#12011yescharacteristic
220C23×C3#12111yescharacteristic
221C23×C3#12711yescharacteristic
222C22×D3—42no—
223C22×D3—42no—
224C22×D3—41no—
225C22×D3—42no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
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
239#67—82no—
240#67—82no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255D12—42no—
256D12—81no—
257Dic6—42no—
258Dic6—81no—
259C3×C8—42no—
260C3×C8—42no—
261C24—61no—
262C24C2×D1211yescharacteristic
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—122no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—242no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—121no—
280C22×C4C2×D1211yescharacteristic
281C22×C4C2×D1211yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4#23911yescharacteristic
286M16—42no—
287M16—42no—
288C2×C8—22no—
289C2×C8—22no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—242no—
301C22⋊C4—42no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—122no—
309C22⋊C4—242no—
310C22⋊C4—242no—
311C42—121no—
312C42—121no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—82no—
322C22×C3#39211yescharacteristic
323C22×C3#45812yes—
324C2×D3—84no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333Dic3—81no—
334Dic3—81no—
335Dic3—81no—
336Dic3—81no—
337Dic3—81no—
338Dic3—161no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—122no—
343C23—122no—
344C23—121no—
345C23—122no—
346C23#87411yescharacteristic
347C23#88211yescharacteristic
348C23#89511yescharacteristic
349Q8—122no—
350Q8—241no—
351D4—122no—
352D4—241no—
353D4—242no—
354D4—242no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4#88211yescharacteristic
379C2×C4#88211yescharacteristic
380C2×C4#89511yescharacteristic
381C2×C4#89511yescharacteristic
382C8—42no—
383C8—42no—
384C2×C3—22no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3#200811yescharacteristic
388C2×C3#200911yescharacteristic
389C2×C3#202211yescharacteristic
390D3—161no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—82no—
400C22—244no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22#467111yescharacteristic
405C22#474212yes—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—241no—
412C4—241no—
413C4—241no—
414C4—241no—
415C4—241no—
416C4—481no—
417C3#998411yescharacteristic, complemented, Sylow
418C2—22no—
419C2—41no—
420C2—41no—
421C2—481no—
422C2#6896611yescharacteristic
423C2#6902811yescharacteristic
424C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1306D3
hypocenterC3#9984
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
1. #78625C2semidirect
4. #69981C2semidirect
6. #69534C2semidirect
7. C3×#1306C2semidirect
40. #1306D3semidirect
417. C3#9984semidirect

Maximal subgroups

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

subgroupindex
#786252
#786242
#741452
#699812
#699782
#695342
C3×#13062
#99843