Finite Group Databasebeta

← #10828028

Subgroups — #10828028

Order 768.

1214 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 187 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#87251C211yesmaximal, characteristic
2C3×#2310C211yesmaximal, characteristic, complemented
3#82216C212yesmaximal, complemented
4#72133C211yesmaximal, characteristic, complemented
5#71591C212yesmaximal, complemented
6#69695C214yesmaximal, complemented
7#69692C212yesmaximal, complemented
8#69672C212yesmaximal, complemented
9#24023—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×#1007C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14#5817—24no—
15#5816—22no—
16#5816C2212yes—
17Q8×#67—24no—
18#5812C2211yescharacteristic
19#5791—22no—
20#5791C2212yes—
21C3×#391—22no—
22C3×#391C2212yes—
23C3×#391C2212yes—
24C3×#381—22no—
25C2×#836C2212yes—
26C2×#836C2212yes—
27#5207C2212yes—
28#5003—24no—
29#4996—24no—
30#4966C2214yes—
31#4964C2214yes—
32C2×#805C2214yes—
33#4523C2214yes—
34#3486—31no—
35#3093—32no—
36#2310D311yescharacteristic, complemented
37#2038—34no—
38#1897—32no—
39#1857—32no—
40#1793—32no—
41#1784—31no—
42C22×C3×Q8—24no—
43C22×C3×Q8—24no—
44C22×C3×Q8—42no—
45C22×C3×Q8D412yes—
46C22×C3×Q8C2311yescharacteristic
47C22×C3×Q8C2311yescharacteristic
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50#1007—24no—
51#1007—42no—
52C2×#239—22no—
53C2×D3×Q8—22no—
54C2×#235—22no—
55C2×#235C2312yes—
56C3×#117—24no—
57C3×#117—42no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C42—21no—
63C2×C3×C42C2312yes—
64#953—22no—
65#947—24no—
66#947—22no—
67#946—28no—
68#946—24no—
69Q8×Dic3—24no—
70#944—24no—
71#944—22no—
72#930—24no—
73#930D414yes—
74#929—24no—
75#929—22no—
76C4×#67—28no—
77C4×#67—24no—
78#926—22no—
79#926C2312yes—
80#924—24no—
81#924D414yes—
82C2×#208C2314yes—
83C3×#95—24no—
84C3×#95—24no—
85#836—24no—
86#836—42no—
87#830—24no—
88#810—24no—
89#809—24no—
90#808—24no—
91#807—24no—
92#805—44no—
93C2×2−1+4—31no—
94C23×Q8C2×D311yescharacteristic
95D4×Q8—64no—
96#548—64no—
97#536—32no—
98#536—62no—
99#525—32no—
100#525—62no—
101#519—31no—
102C2×#117C2×D311yescharacteristic
103C2×#117C2×D311yescharacteristic
104#485—64no—
105#482—64no—
106#428—34no—
107C2×#104—34no—
108#417—34no—
109C2×#101—32no—
110C2×#101—32no—
111#407—32no—
112#391—22no—
113#391C2×D312yes—
114#391C2×D312yes—
115#381—22no—
116#339—34no—
117C24×C3C2×D411yescharacteristic
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—42no—
126C2×C3×Q8—416no—
127C2×C3×Q8#4114yes—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D414yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2411yescharacteristic
140C2×#67—24no—
141C2×#67—22no—
142#239—48no—
143D3×Q8—48no—
144#235—24no—
145#235—24no—
146#235—44no—
147#235—44no—
148C2×D12—22no—
149C2×D12C2×D412yes—
150C2×C4×D3—24no—
151C2×C4×D3—22no—
152C2×Dic6—22no—
153C2×Dic6C2×D412yes—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—42no—
156C3×C22⋊C4—42no—
157C3×C22⋊C4—42no—
158C3×C42—24no—
159C3×C42—42no—
160#217—24no—
161#217—22no—
162(C2×D3)⋊C4—28no—
163(C2×D3)⋊C4—24no—
164C4⋊Dic3—24no—
165C4⋊Dic3C2×D414yes—
166Dic3⋊C4—28no—
167Dic3⋊C4—24no—
168C4×Dic3—24no—
169C4×Dic3—22no—
170#208—24no—
171#208—24no—
172C2×C3⋊C8—24no—
1732−1+4—64no—
1742−1+4—122no—
175C2×#41—32no—
176C2×#41—62no—
177C2×#41—62no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—42no—
181C22×Q8—62no—
182C22×Q8#6712yes—
183C22×Q8C22×D311yescharacteristic
184C22×Q8C22×D311yescharacteristic
185C23×C4#6712yes—
186C23×C4C22×D311yescharacteristic
187C2×M16—34no—
188C4⋊Q8—62no—
189C4⋊Q8—64no—
190#124—62no—
191#124—64no—
192#123—62no—
193#123—64no—
194C22⋊Q8—34no—
195C22⋊Q8—64no—
196C22⋊Q8—64no—
197C22⋊Q8—68no—
198#121—34no—
199#121—64no—
200#120—62no—
201C4×Q8—64no—
202C4×D4—68no—
203C4×D4—64no—
204#117—24no—
205#117—32no—
206#117—42no—
207#117—62no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—21no—
213C2×C42C22×D312yes—
214#107—64no—
215#106—64no—
216#104—124no—
217#103—64no—
218#102—64no—
219#101—64no—
220#101—122no—
221C22⋊C8—64no—
222#95—24no—
223#95—24no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—42no—
227C23×C3#12012yes—
228C23×C3C22×D411yescharacteristic
229C22×D3—22no—
230C3×Q8—28no—
231C3×Q8—48no—
232C3×Q8—44no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—28no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—48no—
242C2×C3×C4—44no—
243C2×C3×C4—44no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4#12012yes—
247C2×C3×C4#12314yes—
248C2×C3×C4C22×D412yes—
249C2×C3×C4C2×#4112yes—
250C2×C3×C4H(5,𝔽2)12yes—
251#67—48no—
252#67—42no—
253#67—42no—
254C2×Dic3—24no—
255C2×Dic3—22no—
256D12—24no—
257D12—44no—
258C4×D3—48no—
259C4×D3—44no—
260Dic6—24no—
261Dic6—44no—
262C3⋊C8—44no—
263C24C2×#6711yescharacteristic
264#41—64no—
265#41—64no—
266#41—124no—
267#41—128no—
268#41—124no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—32no—
277C2×Q8—42no—
278C2×Q8—416no—
279C2×Q8—62no—
280C2×Q8—128no—
281C2×Q8#23514yes—
282C2×D4—32no—
283C2×D4—62no—
284C2×D4—64no—
285C2×D4—62no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—64no—
294C22×C4—62no—
295C22×C4D3×D414yes—
296C22×C4C2×#6712yes—
297C22×C4C2×#6712yes—
298C22×C4C2×#6711yescharacteristic
299C22×C4C23×D311yescharacteristic
300M16—64no—
301M16—64no—
302C2×C8—64no—
303C4⋊C4—24no—
304C4⋊C4—34no—
305C4⋊C4—42no—
306C4⋊C4—64no—
307C4⋊C4—68no—
308C4⋊C4—64no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—68no—
312C22⋊C4—64no—
313C22⋊C4—62no—
314C22⋊C4—64no—
315C42—24no—
316C42—42no—
317C42—64no—
318C42—62no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3C2×#12011yescharacteristic
326C22×C3C2×#12311yescharacteristic
327C22×C3#53411yescharacteristic
328C2×D3—42no—
329C2×D3—44no—
330C3×C4—24no—
331C3×C4—22no—
332C3×C4—42no—
333C3×C4—82no—
334C3×C4—82no—
335C3×C4#54614yes—
336Dic3—42no—
337Dic3—44no—
338C23—21no—
339C23—21no—
340C23—42no—
341C23—62no—
342C23#93712yes—
343C23C22×#6711yescharacteristic
344Q8—28no—
345Q8—48no—
346Q8—44no—
347Q8—64no—
348Q8—124no—
349D4—64no—
350D4—124no—
351D4—128no—
352D4—122no—
353D4—122no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—64no—
368C2×C4—62no—
369C2×C4—128no—
370C2×C4—124no—
371C2×C4#92914yes—
372C2×C4#93712yes—
373C2×C4C2×#23512yes—
374C2×C4C2×D3×D412yes—
375C2×C4#100412yes—
376C8—124no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—42no—
380C2×C3#240511yescharacteristic
381C2×C3#302012yes—
382D3—82no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—122no—
390C22—124no—
391C22C2×#92911yescharacteristic
392C22C2×#93711yescharacteristic
393C22#580411yescharacteristic
394C4—22no—
395C4—24no—
396C4—42no—
397C4—82no—
398C4—82no—
399C4—122no—
400C4—124no—
401C4#549014yes—
402C3#2402311yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—42no—
406C2—242no—
407C2#8212611yescharacteristic
408C2#8341512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC22×C3C2×#120
prime radical#87251C2
hypercenter#2310D3
hypocenterC3#24023
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C3×#2310C2semidirect
3. #82216C2semidirect
4. #72133C2semidirect
5. #71591C2semidirect
6. #69695C2semidirect
7. #69692C2semidirect
8. #69672C2semidirect
36. #2310D3semidirect
402. C3#24023semidirect

Maximal subgroups

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

subgroupindex
#872512
C3×#23102
#822162
#721332
#715912
#696952
#696922
#696722
#240233