Finite Group Databasebeta

← #10697609

Subgroups — #10697609

Order 768.

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

normal subgroups only

This group has 556 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#83350C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic, complemented
3#73802C211yesmaximal, characteristic
4C3×#1503C211yesmaximal, characteristic, complemented
5#70778C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic, complemented
8#10247—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#875—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5087—21no—
16#5084—21no—
17#5080—21no—
18#5079—21no—
19#5066—22no—
20#5043—22no—
21#5020—22no—
22#5001—21no—
23#4995—21no—
24#4987—22no—
25#4929—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4674—22no—
29C3×#328—22no—
30#4486—22no—
31#4459—22no—
32#3229—31no—
33#3077—31no—
34#2547—31no—
35#1546—31no—
36#1503D311yescharacteristic, complemented
37#1478—31no—
38#1303—31no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
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×(C2×D3)⋊C4—24no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D412yes—
58C2×#208—21no—
59#875—42no—
60#875—42no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×#95D412yes—
73#835—24no—
74C3⋊#95—24no—
75#830—21no—
76#809—22no—
77#809—22no—
78#808—22no—
79#808—24no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×#124—61no—
88C2×C4×D4—31no—
89#498—61no—
90#487—61no—
91#483—62no—
92#478—62no—
93#476—61no—
94#475—62no—
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⋊C4C2×D311yescharacteristic
105#332—62no—
106#328—22no—
107D4⋊C8—62no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—22no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12SD1612yes—
123C2×C4×D3—44no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×C3×C8—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148(C2×D3)⋊C4—48no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—21no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C22×Q8—61no—
162C22×D4—31no—
163C23×C4—61no—
164C2×M16—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8D1211yescharacteristic
170C4⋊Q8D1211yescharacteristic
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—32no—
186C2×C4⋊C4—61no—
187C2×C22⋊C4—64no—
188C2×C22⋊C4—61no—
189C2×C42—21no—
190C2×C42C22×D311yescharacteristic
191#107—62no—
192#106—62no—
193#106—64no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198#103—64no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C4×C8—21no—
204C4×C8—61no—
205#95—64no—
206#95#6712yes—
207C23×C3#12011yescharacteristic
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—42no—
211C22×D3—41no—
212C22×D3—41no—
213C3×Q8—42no—
214C3×Q8—81no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237D12—22no—
238D12—22no—
239D12—41no—
240D12—41no—
241C4×D3—82no—
242C4×D3—82no—
243Dic6—81no—
244Dic6—81no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C24—61no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×Q8—121no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×D4—32no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C22×C4—22no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4—124no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4C2×#6711yescharacteristic
274M16—121no—
275M16—121no—
276C2×C8—41no—
277C2×C8—62no—
278C2×C8—62no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—62no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—62no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—41no—
293C22⋊C4—128no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42C2×D1211yescharacteristic
300C42#23511yescharacteristic
301C42D3×D411yescharacteristic
302C42D3×D411yescharacteristic
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—241no—
335Q8—241no—
336Q8—241no—
337Q8—241no—
338D4—62no—
339D4—62no—
340D4—121no—
341D4—121no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—242no—
355C2×C4—242no—
356C2×C4#88211yescharacteristic
357C2×C4#88211yescharacteristic
358C2×C4#88411yescharacteristic
359C2×C4#88411yescharacteristic
360C2×C4D3×SD1611yescharacteristic
361C2×C4#91411yescharacteristic
362C2×C4#91811yescharacteristic
363C2×C4#91911yescharacteristic
364C2×C4#93011yescharacteristic
365C2×C4#94011yescharacteristic
366C8—81no—
367C8—122no—
368C8—122no—
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#496011yescharacteristic
383C22#516911yescharacteristic
384C22#518711yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—81no—
390C4—81no—
391C4—242no—
392C4—241no—
393C4#480611yescharacteristic
394C4#480711yescharacteristic
395C4#481011yescharacteristic
396C4#481211yescharacteristic
397C4#482712yes—
398C3#1024711yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—242no—
402C2#7005311yescharacteristic
403C2#7027211yescharacteristic
404C2#7038811yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1503D3
hypocenterC3#10247
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
2. #83323C2semidirect
4. C3×#1503C2semidirect
5. #70778C2semidirect
7. #67620C2semidirect
36. #1503D3semidirect
398. C3#10247semidirect

Maximal subgroups

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

subgroupindex
#833502
#833232
#738022
C3×#15032
#707782
#707572
#676202
#102473