Finite Group Databasebeta

← #10690468

Subgroups — #10690468

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4694C211yesmaximal, characteristic
2#73651C211yesmaximal, characteristic, complemented
3#70457C211yesmaximal, characteristic
4C2×#4482C211yesmaximal, characteristic, complemented
5#69487C211yesmaximal, characteristic
6C3×#1279C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9062—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C4×Dic6—21no—
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#821—22no—
16C2×#821—21no—
17C2×#821—21no—
18Dic3×C4⋊C4—21no—
19#4972—21no—
20C2×#809—21no—
21C2×#809—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4929—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4694—22no—
26#4694C414yescomplemented
27#4654—22no—
28#4482—22no—
29C2×#486—31no—
30#2365—31no—
31#1982—31no—
32#1982—31no—
33C2×#332—31no—
34#1279—31no—
35#1279D311yescharacteristic, complemented
36C22×D12—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×Dic6—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D412yes—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—22no—
51C2×C4×Dic3—22no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54#874—21no—
55#874—21no—
56#874—21no—
57#874C2×C412yes—
58#869—21no—
59#869—21no—
60#869C2×C412yes—
61#869D412yes—
62C4×Dic6—42no—
63C4×Dic6—42no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4×C8—21no—
67C3×C4×C8C2×C412yes—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72#821—28no—
73#821—42no—
74#821—42no—
75#809—42no—
76#809—42no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C2×C4⋊Q8—31no—
80C2×#127—31no—
81C2×C4×Q8—61no—
82#486—34no—
83#486—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#102—62no—
87C2×#102—61no—
88C2×#102—61no—
89C2×#102—61no—
90C2×#102—61no—
91C2×C4×C8C2×D311yescharacteristic
92#391—61no—
93#389—61no—
94#387—62no—
95C4×C4⋊C4—31no—
96C4×C4⋊C4—61no—
97C4×C4⋊C4C2×D311yescharacteristic
98#332—62no—
99C23×D3—41no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—22no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—42no—
112C2×D12—42no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×Dic6—22no—
125C2×Dic6—22no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4SD1612yes—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3C22⋊C414yes—
153C4⋊Dic3D812yes—
154Dic3⋊C4—44no—
155C4×Dic3—44no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C22×Q8—61no—
162C22×D4—61no—
163C22×D4—61no—
164C22×D4—61no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—61no—
168C4⋊Q8—32no—
169C4⋊Q8—32no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172#127—32no—
173#127—61no—
174#127—61no—
175#127—61no—
176C4×Q8—122no—
177C4×Q8—122no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4D1211yescharacteristic
184C2×C4⋊C4D1211yescharacteristic
185C2×C42—21no—
186C2×C42—62no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#102—68no—
194#102—122no—
195#102—122no—
196#102—122no—
197#102—122no—
198C4×C8—21no—
199C4×C8C4×D312yes—
200#95—21no—
201#95—62no—
202#95—62no—
203#95—61no—
204C23×C3#12011yescharacteristic
205C22×D3—42no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—81no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4C2×C22⋊C411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4C2×D811yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4#13711yescharacteristic
234C2×Dic3—24no—
235C2×Dic3—22no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243D12—44no—
244D12—42no—
245D12—42no—
246D12—42no—
247D12—42no—
248D12—82no—
249D12—81no—
250D12—81no—
251D12—81no—
252D12—81no—
253Dic6—42no—
254Dic6—42no—
255Dic6—42no—
256Dic6—42no—
257C3×C8—42no—
258C3×C8—81no—
259C3⋊C8—81no—
260C24—121no—
261C2×Q8—62no—
262C2×Q8—62no—
263C2×Q8—121no—
264C2×Q8—121no—
265C2×Q8—121no—
266C2×Q8—121no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—122no—
271C2×D4—122no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—121no—
298C2×C8—121no—
299C2×C8—121no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—34no—
304C4⋊C4—32no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—124no—
312C4⋊C4#20412yes—
313C42—41no—
314C42—41no—
315C42—124no—
316C42—121no—
317C42—121no—
318C42C2×C4×D311yescharacteristic
319C42#23511yescharacteristic
320C42D3×D411yescharacteristic
321C42#23911yescharacteristic
322C22×C3#38611yescharacteristic
323C22×C3#44711yescharacteristic
324C22×C3#44811yescharacteristic
325C22×C3#44911yescharacteristic
326C22×C3#44911yescharacteristic
327C22×C3#45411yescharacteristic
328C22×C3#45411yescharacteristic
329C2×D3—82no—
330C2×D3—82no—
331C2×D3—82no—
332C2×D3—161no—
333C2×D3—161no—
334C2×D3—161no—
335C2×D3—161no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—81no—
340C3×C4C4×D811yescharacteristic
341C3×C4C4×SD1611yescharacteristic
342C3×C4#44011yescharacteristic
343C3×C4#44211yescharacteristic
344C3×C4#45911yescharacteristic
345C3×C4#46011yescharacteristic
346C3×C4#46311yescharacteristic
347C3×C4#46411yescharacteristic
348Dic3—44no—
349Dic3—42no—
350Dic3—81no—
351C23—122no—
352C23—241no—
353C23—241no—
354C23—241no—
355C23—241no—
356C23—241no—
357C23—241no—
358C23#88211yescharacteristic
359Q8—122no—
360Q8—122no—
361Q8—122no—
362Q8—122no—
363D4—124no—
364D4—122no—
365D4—122no—
366D4—122no—
367D4—122no—
368D4—242no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—64no—
382C2×C4—62no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4C4×D1211yescharacteristic
391C2×C4D3×C22⋊C411yescharacteristic
392C2×C4#88411yescharacteristic
393C2×C4#89311yescharacteristic
394C2×C4#89411yescharacteristic
395C2×C4#89511yescharacteristic
396C2×C4C2×#20411yescharacteristic
397C2×C4#90411yescharacteristic
398C2×C4#90811yescharacteristic
399C2×C4#90911yescharacteristic
400C2×C4D3×D811yescharacteristic
401C2×C4#91411yescharacteristic
402C2×C4#91411yescharacteristic
403C2×C4#91911yescharacteristic
404C8—42no—
405C8—81no—
406C8—241no—
407C2×C3#162511yescharacteristic
408C2×C3#162611yescharacteristic
409C2×C3#162711yescharacteristic
410C2×C3#162811yescharacteristic
411C2×C3#187911yescharacteristic
412C2×C3#188011yescharacteristic
413C2×C3#188811yescharacteristic
414D3—162no—
415C22—242no—
416C22—242no—
417C22—242no—
418C22—481no—
419C22—481no—
420C22—481no—
421C22—481no—
422C22#466511yescharacteristic
423C22#473411yescharacteristic
424C22#473811yescharacteristic
425C22#477311yescharacteristic
426C22#481011yescharacteristic
427C22#482711yescharacteristic
428C22#482711yescharacteristic
429C4—21no—
430C4—21no—
431C4—42no—
432C4—81no—
433C4—124no—
434C4—122no—
435C4—241no—
436C4C4×#20411yescharacteristic
437C4#471011yescharacteristic
438C4#478611yescharacteristic
439C4#481911yescharacteristic
440C4#484511yescharacteristic
441C4#484811yescharacteristic
442C4#486911yescharacteristic
443C4#487211yescharacteristic
444C3#906211yescharacteristic, complemented, Sylow
445C2—482no—
446C2#6818011yescharacteristic
447C2#6818711yescharacteristic
448C2#6824511yescharacteristic
449C2#6825211yescharacteristic
450C2#6868611yescharacteristic
451C2#6876611yescharacteristic
452C2#6889711yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9062
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #73651C2semidirect
4. C2×#4482C2semidirect
6. C3×#1279C2semidirect
7. #67655C2semidirect
26. #4694C4semidirect
35. #1279D3semidirect
444. C3#9062semidirect

Maximal subgroups

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

subgroupindex
C2×#46942
#736512
#704572
C2×#44822
#694872
C3×#12792
#676552
#90623