Finite Group Databasebeta

← #10698992

Subgroups — #10698992

Order 768.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5027C211yesmaximal, characteristic
2C3×#1984C211yesmaximal, characteristic, complemented
3#71626C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic
5#69772C211yesmaximal, characteristic
6C2×#4458C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#10474—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#215—21no—
11C2×#874C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#835—21no—
15C2×#821—21no—
16C2×#821—21no—
17#5027—22no—
18#5027—22no—
19C2×#809—21no—
20#4935C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×#336—21no—
24C3×#336—21no—
25#4529—21no—
26#4485—22no—
27#4458—22no—
28C2×#460—31no—
29#1984D311yescharacteristic, complemented
30#1957—31no—
31#1803—31no—
32C2×#335—31no—
33C2×#331—31no—
34#1296—31no—
35C22×C3×Q8D411yescharacteristic
36C22×D12—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×C3×C8—21no—
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51C2×#215—42no—
52C2×#215—42no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58#874—21no—
59#874—21no—
60#874—21no—
61#874D412yes—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4×C8—21no—
65C3×C4×C8—21no—
66#835—42no—
67#821—42no—
68#821—42no—
69#809—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C22×SD16—61no—
75C2×#127—31no—
76C2×C4×Q8C2×D311yescharacteristic
77#460—62no—
78#460—62no—
79C2×C4⋊C8—31no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8C2×D311yescharacteristic
82C2×#103—61no—
83C2×#102—61no—
84C2×#102—61no—
85C2×#102—61no—
86C2×C4×C8C2×D311yescharacteristic
87#384—31no—
88#340—61no—
89#336—21no—
90#336—21no—
91#335—62no—
92#331—62no—
93C23×D3—41no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8D812yes—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—41no—
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—81no—
121C2×D12—81no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8SD1612yescomplemented
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42—41no—
136C3×C42—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141#215—48no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—81no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C22×Q8D1211yescharacteristic
154C22×D4—61no—
155C22×D4—61no—
156C22×D4—61no—
157C2×SD16—64no—
158C2×SD16—122no—
159C2×SD16—122no—
160C22×C8—21no—
161C22×C8—61no—
162C22×C8—61no—
163C22×C8#6711yescharacteristic
164C22×C8#6711yescharacteristic
165#127—32no—
166#127—61no—
167#127—61no—
168#127—61no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4D1211yescharacteristic
178C2×C42—21no—
179C2×C42C22×D311yescharacteristic
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—32no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186#103—122no—
187#102—64no—
188#102—122no—
189#102—122no—
190C4×C8—21no—
191C4×C8—21no—
192C23×C3#12111yescharacteristic
193C22×D3—42no—
194C22×D3—81no—
195C22×D3—81no—
196C22×D3—81no—
197C22×D3—81no—
198C22×D3—81no—
199C22×D3—81no—
200C3×Q8—22no—
201C3×Q8—22no—
202C3×Q8—41no—
203C3×Q8—41no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×C3×C4#12411yescharacteristic
218C2×C3×C4C2×D811yescharacteristic
219C2×C3×C4C2×SD1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—81no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229D12—42no—
230D12—42no—
231D12—42no—
232D12—42no—
233D12—82no—
234D12—82no—
235D12—81no—
236D12—81no—
237D12—81no—
238D12—81no—
239C3×C8—22no—
240C3×C8—41no—
241C3×C8—81no—
242C3⋊C8—42no—
243C3⋊C8—81no—
244C24—121no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8D2412yes—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—241no—
266C2×D4—241no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—121no—
270C22×C4C2×D1211yescharacteristic
271C22×C4#23511yescharacteristic
272C22×C4C2×#6711yescharacteristic
273SD16—128no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—121no—
284C2×C8—121no—
285C2×C8—121no—
286C2×C8—121no—
287C2×C8—121no—
288C2×C8#21512yescomplemented
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—241no—
298C42—41no—
299C42—41no—
300C42D3×D411yescharacteristic
301C42D3×D411yescharacteristic
302C42#23711yescharacteristic
303C42#23711yescharacteristic
304C22×C3#39411yescharacteristic
305C22×C3#46011yescharacteristic
306C22×C3#46411yescharacteristic
307C22×C3#46611yescharacteristic
308C22×C3#46811yescharacteristic
309C22×C3#47111yescharacteristic
310C22×C3#47211yescharacteristic
311C2×D3—82no—
312C2×D3—82no—
313C2×D3—82no—
314C2×D3—161no—
315C2×D3—161no—
316C2×D3—161no—
317C2×D3—161no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—42no—
321C3×C4—81no—
322C3×C4#44711yescharacteristic
323C3×C4#45211yescharacteristic
324C3×C4#46111yescharacteristic
325C3×C4#46311yescharacteristic
326C3×C4#46511yescharacteristic
327C3×C4#46811yescharacteristic
328C3×C4#48011yescharacteristic
329C3×C4#48211yescharacteristic
330Dic3—161no—
331C23—122no—
332C23—241no—
333C23—241no—
334C23—241no—
335C23—241no—
336C23—241no—
337C23—241no—
338C23#93011yescharacteristic
339Q8—22no—
340Q8—22no—
341Q8—41no—
342Q8—41no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—242no—
348D4—242no—
349D4—241no—
350D4—241no—
351D4—241no—
352D4—241no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—241no—
362C2×C4—241no—
363C2×C4—241no—
364C2×C4#88211yescharacteristic
365C2×C4#88311yescharacteristic
366C2×C4#88511yescharacteristic
367C2×C4#88611yescharacteristic
368C2×C4C2×D2411yescharacteristic
369C2×C4#90411yescharacteristic
370C2×C4#90811yescharacteristic
371C2×C4#90911yescharacteristic
372C2×C4#93811yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4C2×#21511yescharacteristic
375C2×C4#94211yescharacteristic
376C2×C4#94911yescharacteristic
377C2×C4#95011yescharacteristic
378C8—22no—
379C8—41no—
380C8—81no—
381C8—122no—
382C8—241no—
383C2×C3#166811yescharacteristic
384C2×C3#168211yescharacteristic
385C2×C3#168611yescharacteristic
386C2×C3#169811yescharacteristic
387C2×C3#203311yescharacteristic
388C2×C3#205111yescharacteristic
389C2×C3#205511yescharacteristic
390D3—162no—
391C22—242no—
392C22—242no—
393C22—242no—
394C22—481no—
395C22—481no—
396C22—481no—
397C22—481no—
398C22#496111yescharacteristic
399C22#502711yescharacteristic
400C22#502811yescharacteristic
401C22#511511yescharacteristic
402C22#513411yescharacteristic
403C22#514111yescharacteristic
404C22#514211yescharacteristic
405C4—21no—
406C4—21no—
407C4—42no—
408C4—81no—
409C4—481no—
410C4#473211yescharacteristic
411C4#473311yescharacteristic
412C4#473611yescharacteristic
413C4#473711yescharacteristic
414C4#504811yescharacteristic
415C4#504911yescharacteristic
416C4#505711yescharacteristic
417C4#508311yescharacteristic
418C3#1047411yescharacteristic, complemented, Sylow
419C2—482no—
420C2#6968011yescharacteristic
421C2#6993411yescharacteristic
422C2#7000311yescharacteristic
423C2#7093911yescharacteristic
424C2#7094111yescharacteristic
425C2#7095811yescharacteristic
426C2#7096011yescharacteristic
subgroupG/H
centerC23#930
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1984C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenter#1984D3
hypocenterC3#10474
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. C3×#1984C2semidirect
3. #71626C2semidirect
6. C2×#4458C2semidirect
7. #67702C2semidirect
29. #1984D3semidirect
130. C2×C3×C8SD16semidirect
288. C2×C8#215semidirect
418. C3#10474semidirect

Maximal subgroups

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

subgroupindex
C2×#50272
C3×#19842
#716262
C2×#44852
#697722
C2×#44582
#677022
#104743