Finite Group Databasebeta

← #10697627

Subgroups — #10697627

Order 768.

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

normal subgroups only

This group has 536 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#79272C211yesmaximal, characteristic, complemented
2#79059C211yesmaximal, characteristic
3#73807C211yesmaximal, characteristic
4#70756C211yesmaximal, characteristic
5#70508C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10249—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5131—21no—
16#5121—21no—
17#4987—22no—
18#4939—21no—
19#4936—21no—
20#4925C2211yescharacteristic
21#4850—22no—
22#4847—22no—
23#4846—21no—
24#4842—21no—
25#4722—22no—
26#4682—21no—
27#4681—21no—
28#4673—22no—
29#4486—22no—
30#4483—22no—
31#4452—22no—
32#3230—31no—
33#3077—31no—
34#2563—31no—
35#1549—31no—
36#1503—31no—
37#1478—31no—
38#1303D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C2×C3×M16—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—24no—
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D412yes—
52#924—21no—
53#924—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12—21no—
57C4×D12—21no—
58#870—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×#95D412yes—
73C3⋊#95—24no—
74#830—21no—
75#821—22no—
76#821—22no—
77#818—22no—
78#818—24no—
79#817—22no—
80#816—24no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×D4—31no—
86#500—61no—
87#487—61no—
88#484—62no—
89#478—62no—
90#477—62no—
91#476—61no—
92#474—61no—
93#431—31no—
94#431C2×D311yescharacteristic
95#428—61no—
96#423C2×D311yescharacteristic
97#418—61no—
98#393—62no—
99#390—61no—
100#382—61no—
101#380—62no—
102C4×C4⋊C4C2×D311yescharacteristic
103#332—62no—
104#328—62no—
105D4⋊C8—62no—
106C23×D3—21no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—22no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12Q1612yes—
119C2×C4×D3—44no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—82no—
127C3×M16—41no—
128C3×M16—41no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C3×C4⋊C4—42no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—84no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—41no—
156C22×Q8—61no—
157C22×D4—31no—
158C23×C4—61no—
159C2×M16—21no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×D4—61no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175#117—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—64no—
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—64no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189C4⋊C8D1212yes—
190C4⋊C8D1212yes—
191#103—64no—
192#102—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C4×C8—21no—
197C4×C8—61no—
198#95—64no—
199#95D1212yes—
200C23×C3#12011yescharacteristic
201C22×D3—22no—
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—41no—
205C22×D3—41no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×C3×C4C2×Q1611yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×C3×C4#13711yescharacteristic
221C2×Dic3—44no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227D12—22no—
228D12—22no—
229D12—41no—
230D12—41no—
231C4×D3—82no—
232C4×D3—82no—
233Dic6—42no—
234Dic6—82no—
235Dic6—82no—
236Dic6—81no—
237C3×C8—42no—
238C3×C8—42no—
239C3⋊C8—81no—
240C24—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—242no—
246C2×D4—32no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C22×C4—22no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—124no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4C2×D1211yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261M16—41no—
262M16—41no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—121no—
266C4⋊C4—42no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—244no—
283C42—41no—
284C42—41no—
285C42—121no—
286C42C2×D1211yescharacteristic
287C42C2×D1211yescharacteristic
288C42D3×D411yescharacteristic
289C42#23911yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#39211yescharacteristic
293C22×C3#45111yescharacteristic
294C22×C3#45211yescharacteristic
295C2×D3—44no—
296C2×D3—42no—
297C2×D3—82no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4#45111yescharacteristic
305C3×C4#45211yescharacteristic
306C3×C4#45412yes—
307C3×C4#46711yescharacteristic
308C3×C4#47011yescharacteristic
309Dic3—81no—
310Dic3—81no—
311Dic3—82no—
312Dic3—81no—
313C23—62no—
314C23—122no—
315C23—122no—
316C23—121no—
317C23—121no—
318C23#88211yescharacteristic
319Q8—122no—
320Q8—241no—
321Q8—242no—
322Q8—242no—
323D4—62no—
324D4—62no—
325D4—121no—
326D4—121no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—124no—
333C2×C4—122no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—242no—
339C2×C4—242no—
340C2×C4#87411yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#89511yescharacteristic
344C2×C4#89511yescharacteristic
345C2×C4#89511yescharacteristic
346C2×C4#91211yescharacteristic
347C2×C4#91511yescharacteristic
348C2×C4#91511yescharacteristic
349C2×C4D3×Q1611yescharacteristic
350C8—42no—
351C8—42no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3#200111yescharacteristic
356C2×C3#200311yescharacteristic
357C2×C3#200811yescharacteristic
358D3—82no—
359C22—21no—
360C22—21no—
361C22—124no—
362C22—122no—
363C22—242no—
364C22—241no—
365C22—241no—
366C22#467111yescharacteristic
367C22#478311yescharacteristic
368C22#480911yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—241no—
374C4—241no—
375C4—242no—
376C4—241no—
377C4#478311yescharacteristic
378C4#480911yescharacteristic
379C4#482712yes—
380C4#486711yescharacteristic
381C4#488711yescharacteristic
382C3#1024911yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—242no—
386C2#6876711yescharacteristic
387C2#6888811yescharacteristic
388C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1303D3
hypocenterC3#10249
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
1. #79272C2semidirect
5. #70508C2semidirect
6. #69471C2semidirect
7. C3×#1303C2semidirect
38. #1303D3semidirect
382. C3#10249semidirect

Maximal subgroups

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

subgroupindex
#792722
#790592
#738072
#707562
#705082
#694712
C3×#13032
#102493