Finite Group Databasebeta

← #10849301

Subgroups — #10849301

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×C3×#557C211yesmaximal, characteristic
2C2×#5082C211yesmaximal, characteristic
3#82483C211yesmaximal, characteristic
4C2×#5022C211yesmaximal, characteristic
5#71781C212yesmaximal
6#71781C212yesmaximal
7#71778C211yesmaximal, characteristic
8#71635C211yesmaximal, characteristic
9#71631C211yesmaximal, characteristic
10#71629C212yesmaximal
11#71628C212yesmaximal
12#25657—31nomaximal, Sylow
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—22no—
18C2×C3×#125—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×#870C2211yescharacteristic
24#5230—21no—
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5082—24no—
33#5022—24no—
34#4983—21no—
35#4976—22no—
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4928C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#557D311yescharacteristic
47C2×#491—31no—
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×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8—24no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#125—216no—
65C3×#125—28no—
66C3×C4×Q8—28no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C42—22no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97#870—22no—
98#870—22no—
99#835—22no—
100#835—22no—
101#835—24no—
102#835—24no—
103#835—24no—
104#835—24no—
105#835—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113#557—24no—
114#557—24no—
115C2×C4⋊Q8C2×D312yes—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125—22no—
118C2×#125—21no—
119C2×#125—31no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2×D312yes—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×Q8C2×D311yescharacteristic
124#491—64no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8—31no—
129C2×#103—31no—
130C2×#103—31no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#103—32no—
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—22no—
147C2×C3×Q8—24no—
148C2×C3×Q8—22no—
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D414yes—
157C2×C3×Q8C2×D412yes—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C3×C4⋊C4—28no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4C2×D412yes—
179C3×C42—28no—
180C3×C42—24no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42C2×D412yes—
184C3×C42C2×D412yes—
185C4⋊Dic3—22no—
186C4⋊Dic3—21no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3#4114yes—
191Dic3⋊C4—42no—
192Dic3⋊C4—42no—
193Dic3⋊C4—44no—
194Dic3⋊C4—82no—
195C4×Dic3—81no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C22×Q8#6711yescharacteristic
204C22×Q8C22×D312yes—
205C22×Q8C22×D311yescharacteristic
206C22×C8—32no—
207C22×C8—31no—
208C22×C8—31no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—22no—
212C4⋊Q8#6714yes—
213#125—216no—
214#125—28no—
215#125—62no—
216#125—62no—
217C4×Q8—28no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8#6714yes—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4#6712yes—
238C2×C4⋊C4#6711yescharacteristic
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—61no—
243C2×C42C22×D312yes—
244C2×C42C22×D311yescharacteristic
245#107—62no—
246#106—62no—
247C4⋊C8—62no—
248#103—64no—
249#103—64no—
250#103—64no—
251#103—62no—
252#103—62no—
253#103—64no—
254#103—64no—
255#97—62no—
256#95—61no—
257#95—62no—
258#95—62no—
259C23×C3C22×D411yescharacteristic
260C3×Q8—28no—
261C3×Q8—22no—
262C3×Q8—22no—
263C3×Q8—24no—
264C3×Q8—24no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×Q8—42no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4#13714yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4112yes—
290C2×C3×C42−1+412yes—
291C2×Dic3—24no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C3⋊C8—41no—
300C3⋊C8—41no—
301C3⋊C8—42no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—21no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8C2×#6714yes—
313C2×Q8C2×#6712yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—62no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4C2×#6712yes—
321C22×C4C2×#6712yes—
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6711yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—62no—
328C2×C8—62no—
329C2×C8—62no—
330C2×C8—61no—
331C2×C8—61no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—34no—
342C4⋊C4—62no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—61no—
347C4⋊C4—124no—
348C4⋊C4—122no—
349C4⋊C4—122no—
350C4⋊C4—242no—
351C4⋊C4C2×#6712yes—
352C42—28no—
353C42—24no—
354C42—22no—
355C42—22no—
356C42—241no—
357C42C2×#6712yes—
358C42C2×#6712yes—
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—44no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4#53211yescharacteristic
372C3×C4#53611yescharacteristic
373C3×C4#54712yes—
374Dic3—44no—
375Dic3—81no—
376Dic3—81no—
377C23C22×#6711yescharacteristic
378Q8—28no—
379Q8—24no—
380Q8—24no—
381Q8—22no—
382Q8—22no—
383Q8—42no—
384Q8—42no—
385Q8—42no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—64no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4#94214yes—
411C2×C4#95312yes—
412C2×C4#95312yes—
413C2×C4D3×#4112yes—
414C2×C4#101012yes—
415C2×C4C22×#6711yescharacteristic
416C2×C4C22×#6711yescharacteristic
417C8—122no—
418C8—121no—
419C8—121no—
420C2×C3#243211yescharacteristic
421C2×C3#300411yescharacteristic
422C2×C3#301311yescharacteristic
423C2×C3#337212yes—
424C2×C3#337412yes—
425C22C2×#94212yes—
426C22#583112yes—
427C22#583711yescharacteristic
428C22#583811yescharacteristic
429C22C2×#95311yescharacteristic
430C4—24no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—124no—
438C4—241no—
439C4—241no—
440C4#582412yes—
441C4#582511yescharacteristic
442C4#583211yescharacteristic
443C3#2565711yescharacteristic, complemented, Sylow
444C2#8369212yes—
445C2#8369912yes—
446C2#8405911yescharacteristic
447C2#8409311yescharacteristic
448C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#557D3
hypocenterC3#25657
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

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×C3×#5572
C2×#50822
#824832
C2×#50222
#717812
#717782
#716352
#716312
#716292
#716282
#256573