Finite Group Databasebeta

← #10697616

Subgroups — #10697616

Order 768.

604 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 101 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#83370C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#73800C211yesmaximal, characteristic
4C3×#1503C211yesmaximal, characteristic, complemented
5#70779C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic, complemented
8#10248—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#897—22no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5093—21no—
16#5092—21no—
17#5087—21no—
18#5084—21no—
19#5066—22no—
20#5050—22no—
21#5031—22no—
22#5001—21no—
23#4995—21no—
24#4939—21no—
25#4935—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4677—22no—
29C3×#328—22no—
30#4481—22no—
31#4459—22no—
32#3209—31no—
33#3077—31no—
34#2542—31no—
35#1548—31no—
36#1503D311yescharacteristic, complemented
37#1478—31no—
38#1303—31no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—24no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×#208—21no—
61#897—48no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3×#95D412yes—
74#830—21no—
75#810—24no—
76#809—22no—
77#809—22no—
78#808—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86C2×C22⋊Q8—62no—
87C2×C4×D4—31no—
88#494—61no—
89#487—61no—
90#478—62no—
91#476—61no—
92#474—61no—
93#462—62no—
94#451—62no—
95#431—31no—
96#431C2×D311yescharacteristic
97#428—61no—
98#423—31no—
99#418—61no—
100#399—62no—
101#390—61no—
102#384—61no—
103C4×C4⋊C4C2×D311yescharacteristic
104#333—62no—
105#328—22no—
106D4⋊C8—62no—
107C23×D3—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4—22no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—22no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12Q1612yescomplemented
122C2×C4×D3—44no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×Dic6—22no—
126C2×Dic6—22no—
127C2×Dic6—44no—
128C2×C3×C8—41no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#216—84no—
145(C2×D3)⋊C4—48no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—48no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C22×Q8—32no—
160C22×D4—31no—
161C23×C4—61no—
162C2×Q16—64no—
163C2×M16—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8D1211yescharacteristic
169C4⋊Q8D1211yescharacteristic
170#125—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—128no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4—61no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—64no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C22⋊C4—64no—
187C2×C22⋊C4—61no—
188C2×C42—21no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—62no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196#103—64no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95#6712yes—
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—42no—
207C22×D3—42no—
208C22×D3—41no—
209C22×D3—41no—
210C3×Q8—42no—
211C3×Q8—81no—
212C2×C3×C4—22no—
213C2×C3×C4—24no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×C3×C4C2×Q1611yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×C3×C4#13711yescharacteristic
229C2×Dic3—42no—
230C2×Dic3—44no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233D12—22no—
234D12—22no—
235D12—41no—
236D12—41no—
237C4×D3—82no—
238C4×D3—82no—
239Dic6—44no—
240Dic6—82no—
241C3×C8—81no—
242C3⋊C8—42no—
243C3⋊C8—42no—
244C24—61no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—62no—
248C2×Q8—62no—
249C2×Q8—124no—
250C2×D4—32no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C22×C4—22no—
257C22×C4—62no—
258C22×C4—61no—
259C22×C4—124no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4C2×#6711yescharacteristic
265Q16—244no—
266M16—121no—
267M16—121no—
268C2×C8—41no—
269C2×C8—62no—
270C2×C8—62no—
271C4⋊C4—24no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—128no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—41no—
286C22⋊C4—128no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42C2×D1211yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#39211yescharacteristic
299C22×C3#45111yescharacteristic
300C22×C3#45211yescharacteristic
301C2×D3—44no—
302C2×D3—42no—
303C2×D3—82no—
304C2×D3—81no—
305C2×D3—81no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—44no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4#45111yescharacteristic
313C3×C4#45211yescharacteristic
314C3×C4#45312yes—
315C3×C4#46711yescharacteristic
316C3×C4#47011yescharacteristic
317Dic3—82no—
318Dic3—81no—
319C23—62no—
320C23—122no—
321C23—122no—
322C23—121no—
323C23—121no—
324C23#93711yescharacteristic
325Q8—42no—
326Q8—81no—
327Q8—124no—
328Q8—242no—
329D4—62no—
330D4—62no—
331D4—121no—
332D4—121no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—124no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—242no—
345C2×C4—242no—
346C2×C4#88211yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#91511yescharacteristic
351C2×C4#91611yescharacteristic
352C2×C4D3×Q1611yescharacteristic
353C2×C4#91811yescharacteristic
354C2×C4#93011yescharacteristic
355C2×C4#94011yescharacteristic
356C8—81no—
357C8—122no—
358C8—122no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3#200111yescharacteristic
362C2×C3#200211yescharacteristic
363C2×C3#200911yescharacteristic
364D3—82no—
365C22—21no—
366C22—21no—
367C22—124no—
368C22—122no—
369C22—242no—
370C22—241no—
371C22—241no—
372C22#496011yescharacteristic
373C22#517311yescharacteristic
374C22#518611yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—44no—
379C4—81no—
380C4—81no—
381C4—241no—
382C4—242no—
383C4#480811yescharacteristic
384C4#480911yescharacteristic
385C4#481311yescharacteristic
386C4#481611yescharacteristic
387C4#482212yes—
388C3#1024811yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—242no—
392C2#7005311yescharacteristic
393C2#7027311yescharacteristic
394C2#7038511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1503D3
hypocenterC3#10248
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
1. #83370C2semidirect
4. C3×#1503C2semidirect
5. #70779C2semidirect
7. #67620C2semidirect
36. #1503D3semidirect
121. C2×D12Q16semidirect
388. C3#10248semidirect

Maximal subgroups

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

subgroupindex
#833702
#833502
#738002
C3×#15032
#707792
#707572
#676202
#102483