Finite Group Databasebeta

← #10849308

Subgroups — #10849308

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×#5567C211yesmaximal, characteristic
2C2×C3×#479C211yesmaximal, characteristic
3#83041C211yesmaximal, characteristic
4C2×#5069C211yesmaximal, characteristic
5#71631C211yesmaximal, characteristic
6#70642C212yesmaximal
7#70588C212yesmaximal
8#70473C211yesmaximal, characteristic
9#70456C211yesmaximal, characteristic
10#70284C212yesmaximal
11#70284C212yesmaximal
12#25657—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#944C2212yes—
16#5567—24no—
17#5567—24no—
18C2×#889—22no—
19C2×#887C2212yes—
20C2×#870—21no—
21C2×#869C2211yescharacteristic
22C2×C4×Dic6—21no—
23C2×C4×Dic6C2211yescharacteristic
24C3×#479—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×#835C2212yes—
31C2×#816C2212yes—
32C2×#816C2212yes—
33#5069—24no—
34#4981—21no—
35#4979—21no—
36#4976—22no—
37#4972C2212yes—
38#4972C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C4×C4⋊Dic3C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#491—31no—
48C2×#479D311yescharacteristic
49#2584—31no—
50#2043—32no—
51#2043—32no—
52#1985—31no—
53#1981—31no—
54#1865—32no—
55#1863—32no—
56#1763—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×Dic6D411yescharacteristic
59C22×Dic6C2312yes—
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×#125—22no—
63C3×#125—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8D414yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74#944—24no—
75#944—24no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4D412yes—
86C2×C4×Dic3—22no—
87C2×C4×Dic3—21no—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2312yes—
90#889—216no—
91#887—24no—
92#887—24no—
93#870—28no—
94#869—22no—
95#869D414yes—
96C4×Dic6—28no—
97C4×Dic6—22no—
98C4×Dic6—22no—
99C3×#107—22no—
100C3×#106—22no—
101C3×C4⋊C8—22no—
102C3×#103—22no—
103C3×#103—22no—
104#835—24no—
105C3⋊#95—21no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108#816—24no—
109#816—24no—
110#810—24no—
111#810—24no—
112#803—22no—
113#557—64no—
114#557—64no—
115C2×C4⋊Q8—32no—
116C2×C4⋊Q8—31no—
117C2×#125—61no—
118C2×#125—62no—
119C2×#125C2×D311yescharacteristic
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—61no—
123C2×C4×Q8C2×D311yescharacteristic
124#491—64no—
125#479—24no—
126C2×#107C2×D311yescharacteristic
127C2×#106C2×D311yescharacteristic
128C2×C4⋊C8C2×D311yescharacteristic
129C2×#103—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103C2×D311yescharacteristic
135C2×#103C2×D311yescharacteristic
136C2×#97—31no—
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4—31no—
143#340—32no—
144#336—32no—
145#336—32no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8C2×D412yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3C2×D412yes—
160C22×Dic3C2×D412yes—
161C2×Dic6—22no—
162C2×Dic6—24no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6C2×D414yes—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4C2×D412yes—
182C3×C42—24no—
183C3×C42C2×D412yes—
184C3×C42C2×D412yes—
185C4⋊Dic3—24no—
186C4⋊Dic3—22no—
187C4⋊Dic3—24no—
188C4⋊Dic3—22no—
189C4⋊Dic3#4114yes—
190Dic3⋊C4—28no—
191Dic3⋊C4—28no—
192Dic3⋊C4—28no—
193Dic3⋊C4—24no—
194Dic3⋊C4—44no—
195Dic3⋊C4—82no—
196C4×Dic3—28no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—81no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C22×Q8—31no—
204C22×Q8—32no—
205C22×Q8C22×D311yescharacteristic
206C22×C8—32no—
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209C4⋊Q8—34no—
210C4⋊Q8—64no—
211C4⋊Q8—64no—
212C4⋊Q8—62no—
213#125—22no—
214#125—22no—
215#125—68no—
216#125—616no—
217C4×Q8—22no—
218C4×Q8—68no—
219C4×Q8—64no—
220C4×Q8—64no—
221C4×Q8—62no—
222C4×Q8—62no—
223C4×Q8#6714yes—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C42—21no—
241C2×C42—32no—
242C2×C42—62no—
243C2×C42—61no—
244C2×C42C22×D311yescharacteristic
245#107—22no—
246#106—22no—
247C4⋊C8—22no—
248#103—22no—
249#103—22no—
250#103—64no—
251#103—64no—
252#103—64no—
253#103—64no—
254#103—64no—
255#97—62no—
256#95—62no—
257#95—62no—
258#95—61no—
259C23×C3C22×D411yescharacteristic
260C3×Q8—22no—
261C3×Q8—22no—
262C3×Q8—42no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4#12012yes—
277C2×C3×C4#12012yes—
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4112yes—
281C2×C3×C42−1+412yes—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—24no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3#13714yes—
294Dic6—28no—
295Dic6—24no—
296Dic6—24no—
297Dic6—42no—
298Dic6—42no—
299C3×C8—41no—
300C3×C8—41no—
301C3⋊C8—42no—
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—34no—
306C2×Q8—64no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8C2×#6712yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—62no—
330C2×C8—62no—
331C2×C8—62no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—34no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—68no—
342C4⋊C4—68no—
343C4⋊C4—68no—
344C4⋊C4—64no—
345C4⋊C4—62no—
346C4⋊C4—64no—
347C4⋊C4—64no—
348C4⋊C4—62no—
349C4⋊C4—124no—
350C4⋊C4—242no—
351C4⋊C4C2×#6712yes—
352C42—24no—
353C42—68no—
354C42—62no—
355C42—62no—
356C42—241no—
357C42D3×D412yes—
358C42C2×#6712yes—
359C22×C3C2×#12011yescharacteristic
360C22×C3#54812yes—
361C22×C3C2×#13712yes—
362C22×C3#57711yescharacteristic
363C22×C3#57811yescharacteristic
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—81no—
369C3×C4—81no—
370C3×C4#53211yescharacteristic
371C3×C4#53611yescharacteristic
372C3×C4#54712yes—
373Dic3—24no—
374Dic3—44no—
375Dic3—44no—
376Dic3—42no—
377C23C2×D3×D411yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—42no—
381Q8—68no—
382Q8—64no—
383Q8—64no—
384Q8—122no—
385Q8—122no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—34no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—64no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—64no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4#93712yes—
412C2×C4#93712yes—
413C2×C4C2×D3×D411yescharacteristic
414C2×C4#100712yes—
415C2×C4D3×#4112yes—
416C2×C4C22×#6711yescharacteristic
417C8—41no—
418C8—41no—
419C8—122no—
420C2×C3#243211yescharacteristic
421C2×C3#300411yescharacteristic
422C2×C3#301311yescharacteristic
423C2×C3#337212yes—
424C2×C3#337412yes—
425C22#563312yes—
426C22#577011yescharacteristic
427C22#577111yescharacteristic
428C22D3×#13712yes—
429C22C2×#93711yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—64no—
435C4—81no—
436C4—81no—
437C4—124no—
438C4—124no—
439C4—122no—
440C4#563112yes—
441C4#581611yescharacteristic
442C4#583211yescharacteristic
443C3#2565711yescharacteristic, complemented, Sylow
444C2#8121912yes—
445C2#8122012yes—
446C2#8275911yescharacteristic
447C2#8391311yescharacteristic
448C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#479C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#479D3
hypocenterC3#25657
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

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×#55672
C2×C3×#4792
#830412
C2×#50692
#716312
#706422
#705882
#704732
#704562
#702842
#256573