Finite Group Databasebeta

← #10804446

Subgroups — #10804446

Order 768.

1394 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2#74051C212yesmaximal, complemented
3C3×#1782C211yesmaximal, characteristic, complemented
4#72139C211yesmaximal, characteristic, complemented
5#70030C218yesmaximal, complemented
6#69537C212yesmaximal, complemented
7#21794—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C2211yescharacteristic
10#5832—22no—
11#5832—22no—
12#5832—22no—
13#5832—22no—
14C2×#947C411yescharacteristic, complemented
15C2×#944C411yescharacteristic, complemented
16C2×#940C411yescharacteristic, complemented
17C2×#939C2214yes—
18C2×#936C411yescharacteristic, complemented
19C2×C4×#67C414yescomplemented
20C22×C4×Dic3C2211yescharacteristic
21C2×C3×#99C2214yes—
22C3×#381C2212yes—
23C2×#834C2214yes—
24#5211C2212yes—
25C2×#806C2218yes—
26#4942C2214yes—
27Dic3×C22⋊C4C2214yes—
28#4651—22no—
29#4650—22no—
30#4649—22no—
31#4648—22no—
32C2×#532—31no—
33#2283—32no—
34#1964—38no—
35#1782—32no—
36#1782—31no—
37#1782D311yescharacteristic, complemented
38C2×C3×#41—22no—
39C2×C3×#41—22no—
40C2×C3×#41—44no—
41C22×C3×Q8C2×C411yescharacteristic
42C22×C3×D4C2×C411yescharacteristic
43C22×C3×D4C2312yes—
44C23×C3×C4C2×C412yes—
45C23×C3×C4C2311yescharacteristic
46C22×#67C2×C414yes—
47C22×#67C2312yes—
48C23×Dic3C2312yes—
49C2×#235—22no—
50C2×#235—22no—
51C2×#235—44no—
52C22×D12C2×C411yescharacteristic
53C22×C4×D3C2×C412yes—
54C22×Dic6C2×C411yescharacteristic
55C2×C3×C22⋊C4C2314yes—
56C2×#217C2×C414yes—
57C2×#217C2314yes—
58#947—21no—
59#947—22no—
60#947—21no—
61#944—21no—
62#944—22no—
63#944—21no—
64#940—21no—
65#940—22no—
66#940—21no—
67#939—24no—
68#939—24no—
69#939—24no—
70#939—24no—
71#936—21no—
72#936—22no—
73#936—21no—
74C4×#67—24no—
75C4×#67—24no—
76C4×#67—24no—
77C4×#67—24no—
78C2×(C2×D3)⋊C4C2×C414yes—
79C2×Dic3⋊C4—24no—
80C2×Dic3⋊C4C2×C414yes—
81C2×C4×Dic3—24no—
82C2×C4×Dic3C2×C412yes—
83C2×C4×Dic3C2×C411yescharacteristic
84C2×C4×Dic3C2×C411yescharacteristic
85C2×C4×Dic3D412yes—
86C2×C4×Dic3D412yes—
87C2×C4×Dic3D412yes—
88C2×C4×Dic3D412yes—
89C3×#99—44no—
90C3×#95—22no—
91C3×#95—22no—
92#834—44no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—24no—
96C3⋊#95—24no—
97#806—48no—
98C22×#41—31no—
99C22×#41C2×D311yescharacteristic
100#532—62no—
101#532—62no—
102#532—62no—
103#532—62no—
104C2×C4⋊Q8—31no—
105C2×#127—31no—
106C2×#124—31no—
107C2×#124—31no—
108C2×#121—34no—
109C2×C4×D4—34no—
110C22×C42—31no—
111C2×#99—34no—
112C2×#99—38no—
113C2×#99C2×D314yes—
114#383—62no—
115#382—62no—
116#381—32no—
117#381C2×D312yes—
118#380—34no—
119C4×C4⋊C4—62no—
120C4×C22⋊C4—34no—
121#376—62no—
122C24×C3C22×C412yes—
123C24×C3C2411yescharacteristic
124C23×D3C22×C412yes—
125C3×#41—48no—
126C3×#41—48no—
127C2×C3×Q8—21no—
128C2×C3×Q8—21no—
129C2×C3×Q8—42no—
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—42no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4C22×C412yes—
142C22×C3×C4C22×C411yescharacteristic
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C2×#67—24no—
147C2×#67—28no—
148C2×#67—24no—
149C2×#67—24no—
150C2×#67—48no—
151C22×Dic3—24no—
152C22×Dic3—44no—
153C22×Dic3C22×C412yes—
154C22×Dic3C22×C412yes—
155C22×Dic3C22×C412yes—
156C22×Dic3C2×D412yes—
157C22×Dic3C2×D412yes—
158C22×Dic3C2×D412yes—
159C22×Dic3C2×D412yes—
160#235—48no—
161#235—48no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—42no—
165C2×C4×D3—24no—
166C2×C4×D3—44no—
167C2×Dic6—21no—
168C2×Dic6—21no—
169C2×Dic6—42no—
170C3×C22⋊C4—28no—
171#217—24no—
172#217—24no—
173#217—28no—
174(C2×D3)⋊C4—24no—
175(C2×D3)⋊C4—24no—
176Dic3⋊C4—24no—
177Dic3⋊C4—24no—
178Dic3⋊C4—48no—
179C4×Dic3—48no—
180C4×Dic3C22⋊C412yes—
181C4×Dic3C22⋊C412yes—
182C4×Dic3C22⋊C412yes—
183C4×Dic3C22⋊C412yes—
184C4×Dic3C22⋊C411yescharacteristic
185C4×Dic3C22⋊C412yes—
186C4×Dic3C22⋊C411yescharacteristic
187C4×Dic3C22⋊C411yescharacteristic
188C4×Dic3C22⋊C412yes—
189C4×Dic3C22⋊C411yescharacteristic
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—44no—
193C2×#41—62no—
194C2×#41—62no—
195C2×#41—124no—
196C22×Q8—31no—
197C22×Q8C4×D311yescharacteristic
198C22×D4—32no—
199C22×D4—34no—
200C22×D4—31no—
201C22×D4C4×D311yescharacteristic
202C22×D4C22×D312yes—
203C23×C4—32no—
204C23×C4—32no—
205C23×C4C4×D312yes—
206C23×C4C22×D311yescharacteristic
207C4⋊Q8—61no—
208C4⋊Q8—62no—
209C4⋊Q8—61no—
210#127—61no—
211#127—62no—
212#127—61no—
213#124—61no—
214#124—62no—
215#124—61no—
216#124—61no—
217#124—62no—
218#124—61no—
219#121—64no—
220#121—64no—
221#121—64no—
222#121—64no—
223C4×D4—64no—
224C4×D4—64no—
225C4×D4—64no—
226C4×D4—64no—
227C2×C4⋊C4—34no—
228C2×C4⋊C4—64no—
229C2×C22⋊C4—34no—
230C2×C22⋊C4—34no—
231C2×C22⋊C4—34no—
232C2×C22⋊C4C22×D314yes—
233C2×C42—32no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—32no—
237C2×C42—32no—
238C2×C42—32no—
239C2×C42—32no—
240C2×C42—64no—
241#99—44no—
242#99—128no—
243#99—124no—
244#95—22no—
245#95—22no—
246#95—64no—
247#95—64no—
248#95—62no—
249#95—62no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—42no—
254C23×C3—44no—
255C23×C3C23×C411yescharacteristic
256C23×C3C22×D412yes—
257C23×C3H(5,𝔽2)14yes—
258C22×D3—22no—
259C22×D3—24no—
260C22×D3—44no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—48no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—48no—
274C2×C3×C4—44no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4C2×C22⋊C411yescharacteristic
278C2×C3×C4C2×C22⋊C411yescharacteristic
279C2×C3×C4C2×C22⋊C412yes—
280#67—48no—
281#67—48no—
282#67—48no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—24no—
286C2×Dic3—22no—
287C2×Dic3—24no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—44no—
292C2×Dic3—44no—
293C2×Dic3—44no—
294C2×Dic3C2×C22⋊C412yes—
295C2×Dic3C2×C22⋊C412yes—
296C2×Dic3C2×C22⋊C412yes—
297C2×Dic3C2×C22⋊C412yes—
298D12—42no—
299D12—42no—
300C4×D3—48no—
301Dic6—42no—
302Dic6—42no—
303C24—32no—
304C24C2×C4×D312yes—
305C24C23×D311yescharacteristic
306#41—48no—
307#41—48no—
308#41—128no—
309#41—128no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—42no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—122no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—64no—
323C2×D4—64no—
324C2×D4—64no—
325C2×D4—68no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—122no—
329C2×D4—128no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—32no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—32no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—64no—
343C22×C4—64no—
344C22×C4—124no—
345C22×C4—124no—
346C22×C4C2×C4×D312yes—
347C22×C4C2×C4×D311yescharacteristic
348C22×C4C2×C4×D311yescharacteristic
349C22×C4D3×D412yes—
350C22×C4D3×D412yes—
351C4⋊C4—64no—
352C4⋊C4—64no—
353C4⋊C4—128no—
354C22⋊C4—28no—
355C22⋊C4—64no—
356C22⋊C4—64no—
357C22⋊C4—64no—
358C22⋊C4—64no—
359C22⋊C4—68no—
360C42—32no—
361C42—32no—
362C42—32no—
363C42—32no—
364C42—31no—
365C42—32no—
366C42—31no—
367C42—31no—
368C42—32no—
369C42—31no—
370C42—128no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—44no—
375C22×C3—42no—
376C22×C3—44no—
377C22×C3—82no—
378C22×C3—82no—
379C22×C3C22×C22⋊C411yescharacteristic
380C22×C3#51812yes—
381C22×C3#53514yes—
382C2×D3—42no—
383C2×D3—44no—
384C2×D3—84no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—84no—
390Dic3—22no—
391Dic3—22no—
392Dic3—22no—
393Dic3—42no—
394Dic3—84no—
395Dic3#41014yes—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—42no—
400C23—44no—
401C23—62no—
402C23—64no—
403C23—124no—
404C23C22×C4×D311yescharacteristic
405C23C2×D3×D412yes—
406C23#100414yes—
407Q8—42no—
408Q8—42no—
409Q8—122no—
410Q8—122no—
411D4—42no—
412D4—42no—
413D4—48no—
414D4—122no—
415D4—122no—
416D4—128no—
417D4—128no—
418D4—128no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—32no—
429C2×C4—32no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—62no—
438C2×C4—64no—
439C2×C4—128no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—124no—
444C2×C4—124no—
445C2×C4—124no—
446C2×C4D3×C22⋊C412yes—
447C2×C4D3×C22⋊C411yescharacteristic
448C2×C4D3×C22⋊C411yescharacteristic
449C2×C3—24no—
450C2×C3—22no—
451C2×C3—82no—
452C2×C3#230911yescharacteristic
453C2×C3C2×#41012yes—
454D3—82no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—44no—
459C22—42no—
460C22—44no—
461C22—82no—
462C22—82no—
463C22—122no—
464C22—124no—
465C22—244no—
466C22C2×D3×C22⋊C411yescharacteristic
467C22#548612yes—
468C22#559214yes—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—34no—
473C4—42no—
474C4—62no—
475C4—62no—
476C4—62no—
477C4—84no—
478C4—122no—
479C4—244no—
480C3#2179411yescharacteristic, complemented, Sylow
481C2—24no—
482C2—22no—
483C2—82no—
484C2—242no—
485C2#7381911yescharacteristic
486C2D3×#41012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5832C2
cosocleC24×C3C24
hypercenter#1782D3
hypocenterC3#21794
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #74051C2semidirect
3. C3×#1782C2semidirect
4. #72139C2semidirect
5. #70030C2semidirect
6. #69537C2semidirect
14. C2×#947C4semidirect
15. C2×#944C4semidirect
16. C2×#940C4semidirect
18. C2×#936C4semidirect
19. C2×C4×#67C4semidirect
37. #1782D3semidirect
480. C3#21794semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
#740512
C3×#17822
#721392
#700302
#695372
#217943