Finite Group Databasebeta

← #10849310

Subgroups — #10849310

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×#5721C211yesmaximal, characteristic
2C2×C3×#491C211yesmaximal, characteristic
3#83041C211yesmaximal, characteristic
4C2×#5063C211yesmaximal, characteristic
5#71778C211yesmaximal, characteristic
6#70284C212yesmaximal
7#70284C212yesmaximal
8#70237C212yesmaximal
9#70235C212yesmaximal
10#69838C211yesmaximal, characteristic
11#69813C211yesmaximal, characteristic
12#25657—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×Q8×Dic3C2212yes—
16C2×#944C2212yes—
17#5721—24no—
18#5721—24no—
19C2×#890—21no—
20C2×#889—22no—
21C2×#887—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#491—24no—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#97C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#816C2212yes—
31#5063—24no—
32#4981—21no—
33#4979—21no—
34#4976—22no—
35#4972C2212yes—
36#4972C2212yes—
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#557—31no—
47C2×#491D311yescharacteristic
48C2×#479—31no—
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×Q8C2312yes—
58C22×Dic6D411yescharacteristic
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8D414yes—
63C3×#125—22no—
64C3×#125—22no—
65C2×C3×C4⋊C4—21no—
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×C42C2311yescharacteristic
72Q8×Dic3—24no—
73Q8×Dic3—24no—
74#944—24no—
75#944—24no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4D412yes—
86C2×C4×Dic3—21no—
87C2×C4×Dic3—22no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92#890—28no—
93#889—216no—
94#887—28no—
95C4×Dic6—22no—
96C4×Dic6—22no—
97C4×Dic6—22no—
98C4×Dic6D414yes—
99C3×#103—24no—
100C3×#103—24no—
101C3×#97—22no—
102#835—24no—
103#835—24no—
104C3⋊#95—21no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#816—24no—
108#810—22no—
109#810—22no—
110#808—22no—
111#807—22no—
112C4⋊(C3⋊C8)—22no—
113#557—64no—
114#557—64no—
115C2×C4⋊Q8—32no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125—62no—
118C2×#125—61no—
119C2×#125C2×D311yescharacteristic
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×Q8—31no—
123C2×C4×Q8—61no—
124#491—24no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8—31no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—31no—
133C2×#103—31no—
134C2×#103C2×D312yes—
135C2×#103C2×D312yes—
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#336—32no—
145#336—32no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8C2×D414yes—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D412yes—
163C2×Dic6—22no—
164C2×Dic6—24no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—21no—
168C2×Dic6C2×D412yes—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182C4⋊Dic3—28no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—22no—
186C4⋊Dic3C2×D412yes—
187C4⋊Dic3#4114yes—
188Dic3⋊C4—28no—
189Dic3⋊C4—28no—
190Dic3⋊C4—24no—
191Dic3⋊C4—24no—
192Dic3⋊C4—44no—
193Dic3⋊C4—82no—
194C4×Dic3—28no—
195C4×Dic3—24no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3—81no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C22×Q8—31no—
204C22×Q8—31no—
205C22×Q8C22×D312yes—
206C22×C8—31no—
207C22×C8—31no—
208C22×C8C22×D312yes—
209C4⋊Q8—22no—
210C4⋊Q8—64no—
211C4⋊Q8—64no—
212C4⋊Q8#6714yes—
213#125—22no—
214#125—22no—
215#125—616no—
216#125—68no—
217C4×Q8—34no—
218C4×Q8—68no—
219C4×Q8—64no—
220C4×Q8—64no—
221C4×Q8—62no—
222C4×Q8—62no—
223C4×Q8—62no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C42—32no—
241C2×C42—62no—
242C2×C42—61no—
243C2×C42—61no—
244C2×C42C22×D311yescharacteristic
245#107—62no—
246#106—62no—
247C4⋊C8—62no—
248#103—24no—
249#103—24no—
250#103—62no—
251#103—62no—
252#103—64no—
253#103—64no—
254#103—64no—
255#97—22no—
256#95—62no—
257#95—62no—
258#95—61no—
259C23×C3C22×D411yescharacteristic
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—42no—
263C3×Q8—42no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—24no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4#12012yes—
276C2×C3×C4#12012yes—
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C2×#4112yes—
280C2×C3×C42−1+412yes—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3#13714yes—
295Dic6—28no—
296Dic6—22no—
297Dic6—22no—
298Dic6—42no—
299C3×C8—42no—
300C3⋊C8—41no—
301C3⋊C8—41no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—32no—
308C2×Q8—64no—
309C2×Q8—62no—
310C2×Q8—61no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8C2×#6714yes—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—62no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8—62no—
331C2×C8—62no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—32no—
339C4⋊C4—34no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—68no—
343C4⋊C4—68no—
344C4⋊C4—68no—
345C4⋊C4—64no—
346C4⋊C4—62no—
347C4⋊C4—64no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—124no—
351C4⋊C4—242no—
352C42—68no—
353C42—64no—
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—44no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4#53211yescharacteristic
370C3×C4#53611yescharacteristic
371C3×C4#54712yes—
372Dic3—24no—
373Dic3—44no—
374Dic3—42no—
375Dic3—42no—
376Dic3—42no—
377C23C2×D3×D411yescharacteristic
378Q8—24no—
379Q8—24no—
380Q8—42no—
381Q8—42no—
382Q8—68no—
383Q8—62no—
384Q8—62no—
385Q8—122no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—34no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—64no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4#93712yes—
412C2×C4#93712yes—
413C2×C4C2×D3×D411yescharacteristic
414C2×C4D3×#4112yes—
415C2×C4#101012yes—
416C2×C4C22×#6711yescharacteristic
417C8—42no—
418C8—121no—
419C8—121no—
420C2×C3#243211yescharacteristic
421C2×C3#300411yescharacteristic
422C2×C3#301311yescharacteristic
423C2×C3#337212yes—
424C2×C3#337412yes—
425C22#567712yes—
426C22D3×#13712yes—
427C22#577811yescharacteristic
428C22#577911yescharacteristic
429C22C2×#93711yescharacteristic
430C4—22no—
431C4—22no—
432C4—44no—
433C4—64no—
434C4—81no—
435C4—81no—
436C4—122no—
437C4—122no—
438C4—122no—
439C4—124no—
440C4#566811yescharacteristic
441C4#567511yescharacteristic
442C4#582412yes—
443C3#2565711yescharacteristic, complemented, Sylow
444C2#8173212yes—
445C2#8173312yes—
446C2#8295911yescharacteristic
447C2#8397911yescharacteristic
448C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#491C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#491D3
hypocenterC3#25657
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

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×#57212
C2×C3×#4912
#830412
C2×#50632
#717782
#702842
#702372
#702352
#698382
#698132
#256573