Finite Group Databasebeta

← #10697190

Subgroups — #10697190

Order 768.

508 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 103 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#79313C211yesmaximal, characteristic, complemented
2#79059C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic
4#70783C211yesmaximal, characteristic
5#70508C211yesmaximal, characteristic, complemented
6#70429C211yesmaximal, characteristic, complemented
7C3×#1301C211yesmaximal, characteristic, complemented
8#10182—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5131—21no—
16#5121—21no—
17C2×#821—21no—
18C2×#817—21no—
19#4925C2211yescharacteristic
20#4849—22no—
21#4847—22no—
22#4846—21no—
23#4842—22no—
24#4842—21no—
25#4842—22no—
26#4739—22no—
27#4699—21no—
28#4681—21no—
29#4486—22no—
30#4484—22no—
31#4468—22no—
32#3209—31no—
33#3077—31no—
34C2×#476—31no—
35#1549—31no—
36#1481—31no—
37#1460—31no—
38#1301D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C2×C3×C42C2311yescharacteristic
45C2×(C2×D3)⋊C4—21no—
46#926—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50#924—21no—
51#924—21no—
52C2×Dic12—24no—
53C4×D12—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73#830—21no—
74#821—22no—
75#821—22no—
76#821—24no—
77#818—22no—
78#817—22no—
79#817—24no—
80#816—24no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×D4—31no—
86#494—61no—
87#487—61no—
88#478—62no—
89#476—61no—
90#476—62no—
91#476—62no—
92#474—61no—
93#462—62no—
94#451—62no—
95#431—31no—
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#106—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8C2×D311yescharacteristic
101#418—61no—
102C2×#102—61no—
103#332—62no—
104Q8⋊C8—62no—
105#323—62no—
106C23×D3—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—41no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12Q1612yescomplemented
118C2×C4×D3—41no—
119C2×C4×D3—41no—
120C2×C4×D3—82no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—41no—
125C2×Dic6—82no—
126C3×M16—41no—
127C3×M16—41no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—41no—
154Dic12—84no—
155C22×Q8—61no—
156C22×D4—31no—
157C23×C4—61no—
158C2×Q16—64no—
159C2×M16—21no—
160C22×C8—21no—
161C4⋊Q8—32no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167#125—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176#117—61no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—64no—
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189C4⋊C8D1212yes—
190C4⋊C8D1212yes—
191#103—64no—
192#102—64no—
193#102—62no—
194#102—62no—
195C22⋊C8—61no—
196C22⋊C8D1212yes—
197C4×C8—21no—
198C4×C8—61no—
199C23×C3#12111yescharacteristic
200C22×D3—22no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—81no—
204C22×D3—81no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×C3×C4C2×Q1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×C3×C4#13711yescharacteristic
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—82no—
222C2×Dic3—81no—
223D12—22no—
224D12—22no—
225D12—41no—
226D12—41no—
227C4×D3—82no—
228C4×D3—82no—
229Dic6—42no—
230Dic6—44no—
231Dic6—42no—
232Dic6—82no—
233C3×C8—42no—
234C3×C8—42no—
235C3×C8—42no—
236C3⋊C8—81no—
237C24—61no—
238C2×Q8—62no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—121no—
242C2×Q8—242no—
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4—242no—
254C22×C4C2×D1211yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4#23911yescharacteristic
257Q16—244no—
258M16—41no—
259M16—41no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—121no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C42—121no—
282C42C2×D1211yescharacteristic
283C42C2×D1211yescharacteristic
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3#39211yescharacteristic
289C22×C3#46711yescharacteristic
290C22×C3#47011yescharacteristic
291C2×D3—42no—
292C2×D3—82no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—82no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4#45111yescharacteristic
300C3×C4#45111yescharacteristic
301C3×C4#45211yescharacteristic
302C3×C4#45211yescharacteristic
303C3×C4#47112yes—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308C23—62no—
309C23—121no—
310C23—121no—
311C23—241no—
312C23—241no—
313C23#89511yescharacteristic
314Q8—122no—
315Q8—124no—
316Q8—122no—
317Q8—242no—
318D4—62no—
319D4—62no—
320D4—121no—
321D4—121no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—122no—
328C2×C4—242no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4—241no—
332C2×C4#87411yescharacteristic
333C2×C4#88211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#89511yescharacteristic
337C2×C4#89511yescharacteristic
338C2×C4#91211yescharacteristic
339C2×C4#91511yescharacteristic
340C2×C4#91511yescharacteristic
341C2×C4D3×Q1611yescharacteristic
342C8—42no—
343C8—42no—
344C8—42no—
345C8—241no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#200111yescharacteristic
349C2×C3#200811yescharacteristic
350C2×C3#201911yescharacteristic
351D3—82no—
352C22—21no—
353C22—21no—
354C22—122no—
355C22—242no—
356C22—242no—
357C22—241no—
358C22—241no—
359C22#467111yescharacteristic
360C22#486711yescharacteristic
361C22#488711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—241no—
366C4—241no—
367C4—242no—
368C4—241no—
369C4#478311yescharacteristic
370C4#478311yescharacteristic
371C4#480911yescharacteristic
372C4#480911yescharacteristic
373C4#489812yes—
374C3#1018211yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—242no—
378C2#6888811yescharacteristic
379C2#6894311yescharacteristic
380C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1301D3
hypocenterC3#10182
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
1. #79313C2semidirect
5. #70508C2semidirect
6. #70429C2semidirect
7. C3×#1301C2semidirect
38. #1301D3semidirect
117. C2×D12Q16semidirect
374. C3#10182semidirect

Maximal subgroups

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

subgroupindex
#793132
#790592
C2×#48422
#707832
#705082
#704292
C3×#13012
#101823