Finite Group Databasebeta

← #10849305

Subgroups — #10849305

Order 768.

980 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic
2#82483C211yesmaximal, characteristic
3C2×#4829C211yesmaximal, characteristic
4C2×#4703C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic
6#70456C211yesmaximal, characteristic
7#69813C211yesmaximal, characteristic
8#68684C212yesmaximal
9#68683C212yesmaximal
10#68631C212yesmaximal
11#68631C212yesmaximal
12#25657—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×Q8×Dic3—21no—
15#5570—24no—
16#5570—24no—
17C2×#890—22no—
18C2×#890—21no—
19C2×#870C2211yescharacteristic
20C2×#869C2212yes—
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2212yes—
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26#5230—21no—
27C2×#818C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#816C2212yes—
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4983—21no—
37#4976—22no—
38C4×C4⋊Dic3C2211yescharacteristic
39#4928C2212yes—
40#4928C2212yes—
41#4829—24no—
42#4703—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#557—31no—
47C2×#491—31no—
48C2×#479—31no—
49#2584—31no—
50#2043—32no—
51#2043—32no—
52#1985D311yescharacteristic
53#1981—31no—
54#1865—32no—
55#1863—32no—
56#1763—31no—
57C22×C3×Q8D411yescharacteristic
58C22×Dic6C2312yes—
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C2×C3×C4⋊C4D412yes—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C42C2312yes—
68C2×C3×C42C2311yescharacteristic
69Q8×Dic3—28no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3C2311yescharacteristic
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—21no—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3—21no—
87#890—216no—
88#890—28no—
89#870—22no—
90#870—22no—
91#869—24no—
92#869—24no—
93#869—22no—
94#869D414yes—
95C4×Dic6—24no—
96C4×Dic6—22no—
97C4×Dic6—24no—
98C4×Dic6D414yes—
99C3×C4⋊C8—22no—
100C3×#97—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—21no—
104#818—22no—
105#817—22no—
106#816—24no—
107#816—24no—
108#816—22no—
109#816—22no—
110#816—24no—
111#816—24no—
112#816—24no—
113#557—64no—
114#557—64no—
115C2×C4⋊Q8—31no—
116C2×C4⋊Q8—32no—
117C2×#125—31no—
118C2×#125—62no—
119C2×#125—61no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—61no—
123C2×C4×Q8C2×D311yescharacteristic
124#491—64no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8C2×D311yescharacteristic
129C2×#103—31no—
130C2×#103—31no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#97C2×D311yescharacteristic
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4—31no—
143#340—32no—
144#336C2×D312yes—
145#336C2×D312yes—
146C2×C3×Q8—22no—
147C2×C3×Q8—24no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2411yescharacteristic
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156C22×Dic3—21no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—21no—
167C2×Dic6C2×D414yes—
168C2×Dic6C2×D412yes—
169C2×C3×C8—22no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182C4⋊Dic3—28no—
183C4⋊Dic3—28no—
184C4⋊Dic3—24no—
185C4⋊Dic3—24no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3C2×D412yes—
193C4⋊Dic3#4114yes—
194Dic3⋊C4—28no—
195Dic3⋊C4—24no—
196Dic3⋊C4—42no—
197Dic3⋊C4—42no—
198Dic3⋊C4—44no—
199Dic3⋊C4—82no—
200C4×Dic3—28no—
201C4×Dic3—24no—
202C4×Dic3—81no—
203C22×Q8—31no—
204C22×Q8—32no—
205C22×Q8D1211yescharacteristic
206C22×C8C22×D312yes—
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209C4⋊Q8—34no—
210C4⋊Q8—64no—
211C4⋊Q8—64no—
212C4⋊Q8—62no—
213#125—616no—
214#125—68no—
215#125—62no—
216#125—62no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—34no—
220C4×Q8—64no—
221C4×Q8—64no—
222C4×Q8—68no—
223C4×Q8—62no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—32no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4D1212yes—
239C2×C4⋊C4D1211yescharacteristic
240C2×C42—62no—
241C2×C42—61no—
242C2×C42—61no—
243C2×C42C22×D312yes—
244C2×C42C22×D311yescharacteristic
245#107—62no—
246#106—62no—
247C4⋊C8—22no—
248#103—64no—
249#103—64no—
250#103—64no—
251#103—62no—
252#103—62no—
253#103—64no—
254#103—64no—
255#97—22no—
256#95—61no—
257#95—62no—
258#95—62no—
259C23×C3C22×D411yescharacteristic
260C3×Q8—28no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4#12012yes—
270C2×C3×C4#12012yes—
271C2×C3×C4#13714yes—
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C2×#4112yes—
275C2×C3×C42−1+412yes—
276C2×Dic3—22no—
277C2×Dic3—24no—
278C2×Dic3—22no—
279C2×Dic3—24no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292Dic6—22no—
293Dic6—22no—
294Dic6—24no—
295Dic6—24no—
296Dic6—42no—
297Dic6—42no—
298Dic6—42no—
299C3×C8—41no—
300C3×C8—41no—
301C3×C8—42no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—32no—
305C2×Q8—34no—
306C2×Q8—62no—
307C2×Q8—61no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C22×C4—62no—
315C22×C4—61no—
316C22×C4—62no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4C2×D1212yes—
321C22×C4C2×D1211yescharacteristic
322C22×C4C2×D1211yescharacteristic
323C22×C4C2×D1212yes—
324C22×C4C23×D311yescharacteristic
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—32no—
335C4⋊C4—34no—
336C4⋊C4—68no—
337C4⋊C4—68no—
338C4⋊C4—68no—
339C4⋊C4—64no—
340C4⋊C4—64no—
341C4⋊C4—64no—
342C4⋊C4—61no—
343C4⋊C4—62no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—122no—
349C4⋊C4—122no—
350C4⋊C4—124no—
351C4⋊C4—242no—
352C42—22no—
353C42—22no—
354C42—68no—
355C42—64no—
356C42—241no—
357C42D3×D412yes—
358C42D3×D412yes—
359C22×C3C2×#12011yescharacteristic
360C22×C3#54812yes—
361C22×C3C2×#13712yes—
362C22×C3#57711yescharacteristic
363C22×C3#57811yescharacteristic
364C3×C4—22no—
365C3×C4—24no—
366C3×C4—22no—
367C3×C4—42no—
368C3×C4#53211yescharacteristic
369C3×C4#53611yescharacteristic
370C3×C4#54712yes—
371Dic3—44no—
372Dic3—42no—
373Dic3—44no—
374Dic3—42no—
375Dic3—81no—
376Dic3—81no—
377C23C22×D1211yescharacteristic
378Q8—28no—
379Q8—62no—
380Q8—62no—
381Q8—64no—
382Q8—64no—
383Q8—122no—
384Q8—122no—
385Q8—122no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—62no—
395C2×C4—64no—
396C2×C4—62no—
397C2×C4—64no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4#88212yes—
411C2×C4#88212yes—
412C2×C4#90914yes—
413C2×C4C2×D3×D411yescharacteristic
414C2×C4C2×D3×D411yescharacteristic
415C2×C4#100712yes—
416C2×C4D3×#4112yes—
417C8—41no—
418C8—41no—
419C8—42no—
420C2×C3#243211yescharacteristic
421C2×C3#300411yescharacteristic
422C2×C3#301311yescharacteristic
423C2×C3#337212yes—
424C2×C3#337412yes—
425C22C2×#88211yescharacteristic
426C22#557612yes—
427C22C2×#90912yes—
428C22#575111yescharacteristic
429C22#575311yescharacteristic
430C4—22no—
431C4—24no—
432C4—22no—
433C4—42no—
434C4—122no—
435C4—124no—
436C4—122no—
437C4—124no—
438C4—241no—
439C4—241no—
440C4#563112yes—
441C4#566811yescharacteristic
442C4#572411yescharacteristic
443C3#2565711yescharacteristic, complemented, Sylow
444C2#7649811yescharacteristic
445C2#7864511yescharacteristic
446C2#7868011yescharacteristic
447C2#7997212yes—
448C2#7998312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socle, prime radicalC23×C3C22×D4
hypercenter#1985D3
hypocenterC3#25657
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
443. C3#25657semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
#824832
C2×#48292
C2×#47032
C3×#19852
#704562
#698132
#686842
#686832
#686312
#256573