Finite Group Databasebeta

← #10698675

Subgroups — #10698675

Order 768.

460 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 91 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#83071C211yesmaximal, characteristic
2C2×C3×#330C211yesmaximal, characteristic, complemented
3#71741C211yesmaximal, characteristic, complemented
4#69821C211yesmaximal, characteristic
5#69816C211yesmaximal, characteristic, complemented
6C2×#4464C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic, complemented
8#10427—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×#821—21no—
14C2×#818—21no—
15C2×#817—21no—
16C2×#816—21no—
17#4994—21no—
18#4994—21no—
19#4993—21no—
20#4992—21no—
21#4991—21no—
22#4979—21no—
23#4936C2211yescharacteristic
24#4935C2211yescharacteristic
25C2×#803C2211yescharacteristic
26C3×#330—22no—
27#4529—21no—
28#4529—21no—
29#4464—22no—
30#2820—31no—
31#1992—31no—
32#1959—31no—
33#1847—31no—
34C2×#330D311yescharacteristic, complemented
35C2×#329—31no—
36#1295—31no—
37C22×D12D411yescharacteristic
38C22×Dic6D411yescharacteristic
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
44C3×#125—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C22×C3⋊C8—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#875—21no—
68C3×#97—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76#821—42no—
77#818—42no—
78#817—42no—
79#816—42no—
80#803—21no—
81#803—21no—
82C2×#125C2×D311yescharacteristic
83C2×#124—31no—
84C2×#107—61no—
85C2×#106—61no—
86C2×#103—61no—
87C2×#102—61no—
88C2×#97—31no—
89C2×#97C2×D311yescharacteristic
90C2×#97C2×D311yescharacteristic
91#399—61no—
92#397—61no—
93#396—61no—
94#396—61no—
95#395—61no—
96#394—61no—
97#384—31no—
98#382—31no—
99#340—61no—
100#340—61no—
101#330—22no—
102#329—62no—
103C23×D3—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—41no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—41no—
116C2×D12—41no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—81no—
144(C2×D3)⋊C4—81no—
145(C2×D3)⋊C4—81no—
146(C2×D3)⋊C4—81no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4⋊Dic3—42no—
154Dic3⋊C4—81no—
155Dic3⋊C4—81no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—31no—
160C22×D4—31no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—61no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168#124—61no—
169#124—61no—
170#124—61no—
171#124—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4D1211yescharacteristic
182C2×C4⋊C4D1211yescharacteristic
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C42C22×D311yescharacteristic
190#107—122no—
191#106—122no—
192#103—122no—
193#102—122no—
194#97—21no—
195#97—21no—
196#97—21no—
197#97—61no—
198#97—61no—
199#97#6712yes—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204C23×C3#12011yescharacteristic
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4#12411yescharacteristic
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242D12—42no—
243D12—81no—
244Dic6—42no—
245Dic6—81no—
246C3×C8—24no—
247C3×C8—42no—
248C3⋊C8—42no—
249C24—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—121no—
257C2×D4—121no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—121no—
264C22×C4C2×D1211yescharacteristic
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—62no—
274C2×C8—121no—
275C2×C8—121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—122no—
291C4⋊C4—241no—
292C4⋊C4—241no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C22⋊C4—121no—
297C22⋊C4—241no—
298C22⋊C4—241no—
299C22⋊C4—241no—
300C22⋊C4—241no—
301C42#23511yescharacteristic
302C42#23711yescharacteristic
303C42#23711yescharacteristic
304C42C2×#6711yescharacteristic
305C22×C3#39411yescharacteristic
306C22×C3#45311yescharacteristic
307C22×C3#45311yescharacteristic
308C22×C3#45411yescharacteristic
309C22×C3#45511yescharacteristic
310C22×C3#45611yescharacteristic
311C22×C3#45611yescharacteristic
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—81no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326C3×C4—81no—
327Dic3—81no—
328Dic3—81no—
329Dic3—81no—
330Dic3—161no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23#88211yescharacteristic
339Q8—122no—
340Q8—241no—
341D4—122no—
342D4—241no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4#88311yescharacteristic
368C2×C4#88511yescharacteristic
369C2×C4#88611yescharacteristic
370C2×C4#93011yescharacteristic
371C2×C4#93811yescharacteristic
372C2×C4#93911yescharacteristic
373C8—24no—
374C8—42no—
375C8—122no—
376C2×C3#165611yescharacteristic
377C2×C3#166011yescharacteristic
378C2×C3#166111yescharacteristic
379C2×C3#166411yescharacteristic
380C2×C3#203711yescharacteristic
381C2×C3#203811yescharacteristic
382C2×C3#203911yescharacteristic
383D3—161no—
384C22—241no—
385C22—241no—
386C22—241no—
387C22—241no—
388C22—241no—
389C22—241no—
390C22—241no—
391C22#475211yescharacteristic
392C22#475311yescharacteristic
393C22#482211yescharacteristic
394C22#482211yescharacteristic
395C22#482511yescharacteristic
396C22#482611yescharacteristic
397C22#496111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—81no—
406C4—241no—
407C4—241no—
408C4—241no—
409C4—481no—
410C3#1042711yescharacteristic, complemented, Sylow
411C2—481no—
412C2#6837011yescharacteristic
413C2#6837211yescharacteristic
414C2#6837311yescharacteristic
415C2#6837911yescharacteristic
416C2#7010011yescharacteristic
417C2#7035911yescharacteristic
418C2#7036111yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#330D3
hypocenterC3#10427
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
2. C2×C3×#330C2semidirect
3. #71741C2semidirect
5. #69816C2semidirect
7. #67659C2semidirect
34. C2×#330D3semidirect
410. C3#10427semidirect

Maximal subgroups

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

subgroupindex
#830712
C2×C3×#3302
#717412
#698212
#698162
C2×#44642
#676592
#104273