Finite Group Databasebeta

← #10697608

Subgroups — #10697608

Order 768.

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

normal subgroups only

This group has 556 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
1C3×#3229C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#83240C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#70756C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic, complemented
8#10247—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#947—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#498—21no—
14C3×#487—21no—
15C3×#483—22no—
16C3×#475—22no—
17C3×#431C2211yescharacteristic
18#5229—21no—
19#5087—21no—
20#5084—21no—
21#5066—22no—
22#5001—21no—
23#4995—21no—
24#4992—22no—
25#4987—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4538—22no—
31#4483—22no—
32#3229D311yescharacteristic, complemented
33#3077—31no—
34#2547—31no—
35#1546—31no—
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C42C2311yescharacteristic
57#947—42no—
58#947—42no—
59C2×(C2×D3)⋊C4—24no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C3×#106—24no—
70C3×C4⋊C8D412yes—
71C3×#103—24no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74C3⋊#95—24no—
75C3⋊#95D412yes—
76#830—21no—
77#809—22no—
78#809—22no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×#124—61no—
88C2×C4×D4—31no—
89#498—21no—
90#487—21no—
91#483—22no—
92#478—62no—
93#476—61no—
94#475—22no—
95#474—61no—
96#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423—31no—
100#418—61no—
101#397—62no—
102#391—61no—
103#380—62no—
104C4×C4⋊C4—31no—
105#332—62no—
106#328—62no—
107D4⋊C8—62no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C22×C3×C4—22no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—22no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12SD1612yes—
129C2×C4×D3—44no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149(C2×D3)⋊C4—48no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—42no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C22×Q8—21no—
162C22×D4—31no—
163C23×C4—61no—
164C2×M16—61no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#125—21no—
172#124—122no—
173#124—122no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6712yes—
187C2×C22⋊C4—64no—
188C2×C22⋊C4—61no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191#107—62no—
192#106—24no—
193#106—62no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8#6712yes—
198#103—24no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C4×C8—21no—
204C4×C8—61no—
205#95—32no—
206#95—64no—
207C23×C3#12011yescharacteristic
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—41no—
212C22×D3—41no—
213C3×Q8—42no—
214C3×Q8—81no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×Q8—81no—
218C3×Q8—81no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×SD1611yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241D12—22no—
242D12—22no—
243D12—41no—
244D12—41no—
245C4×D3—82no—
246C4×D3—82no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250C24—61no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—32no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C22×C4—22no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—124no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4C2×#6711yescharacteristic
274M16—121no—
275M16—121no—
276C2×C8—22no—
277C2×C8—62no—
278C2×C8—121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—122no—
292C22⋊C4—41no—
293C22⋊C4—128no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—41no—
297C42—121no—
298C42—121no—
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C42C2×#6711yescharacteristic
302C42C2×#6711yescharacteristic
303C22×C3—21no—
304C22×C3—21no—
305C22×C3#39211yescharacteristic
306C22×C3#45011yescharacteristic
307C22×C3#45211yescharacteristic
308C2×D3—44no—
309C2×D3—42no—
310C2×D3—82no—
311C2×D3—81no—
312C2×D3—81no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—82no—
317C3×C4—81no—
318C3×C4—81no—
319C3×C4#44811yescharacteristic
320C3×C4#44911yescharacteristic
321C3×C4#45412yes—
322C3×C4#46511yescharacteristic
323C3×C4#46811yescharacteristic
324Dic3—82no—
325Dic3—81no—
326C23—62no—
327C23—122no—
328C23—122no—
329C23—121no—
330C23—121no—
331C23#93711yescharacteristic
332Q8—42no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338D4—62no—
339D4—62no—
340D4—121no—
341D4—121no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—121no—
354C2×C4—242no—
355C2×C4—242no—
356C2×C4D3×SD1611yescharacteristic
357C2×C4#91411yescharacteristic
358C2×C4#91511yescharacteristic
359C2×C4#91611yescharacteristic
360C2×C4#93711yescharacteristic
361C2×C4#93811yescharacteristic
362C2×C4#93911yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#94011yescharacteristic
365C2×C4#95311yescharacteristic
366C8—42no—
367C8—122no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3#200011yescharacteristic
372C2×C3#200311yescharacteristic
373C2×C3#200911yescharacteristic
374D3—82no—
375C22—21no—
376C22—21no—
377C22—124no—
378C22—122no—
379C22—242no—
380C22—241no—
381C22—241no—
382C22#516911yescharacteristic
383C22#517311yescharacteristic
384C22#522311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—81no—
390C4—81no—
391C4—242no—
392C4—241no—
393C4#517011yescharacteristic
394C4#517111yescharacteristic
395C4#517211yescharacteristic
396C4#517411yescharacteristic
397C4#524512yes—
398C3#1024711yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—242no—
402C2#7186711yescharacteristic
403C2#7205411yescharacteristic
404C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3229D3
hypocenterC3#10247
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C3×#3229C2semidirect
4. #71835C2semidirect
5. #71773C2semidirect
7. #67746C2semidirect
32. #3229D3semidirect
398. C3#10247semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833502
#832402
#718352
#717732
#707562
#677462
#102473