Finite Group Databasebeta

← #10814357

Subgroups — #10814357

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82463C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3#74228C211yesmaximal, characteristic
4C3×#1805C211yesmaximal, characteristic, complemented
5#70480C211yesmaximal, characteristic
6#70466C211yesmaximal, characteristic, complemented
7#70276C212yesmaximal
8#70256C212yesmaximal, complemented
9#69547C211yesmaximal, characteristic, complemented
10#68788C212yesmaximal, complemented
11#68782C212yesmaximal
12#22760—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#215C2211yescharacteristic
15C2×#893C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#103C2212yes—
19C3×#382C2211yescharacteristic
20#5229C2212yes—
21C2×#835C2211yescharacteristic
22C2×#821C2212yes—
23C2×#815C2212yes—
24C4×#215C418yescomplemented
25#4980—22no—
26Dic3×C4⋊C4C2212yes—
27#4972C2212yes—
28C2×#809C2211yescharacteristic
29C2×#808C2211yescharacteristic
30#4937—21no—
31#4936C2211yescharacteristic
32C4×C4⋊Dic3C2211yescharacteristic
33#4932—21no—
34C2×C4×C3⋊C8C2211yescharacteristic
35#4668—22no—
36#4667—21no—
37D3×#95—21no—
38C3×#340C2212yes—
39#4551C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×SD16—31no—
44#2356—31no—
45#2349—31no—
46#1935—32no—
47#1919—32no—
48#1877—32no—
49#1835—31no—
50#1805D311yescharacteristic, complemented
51#1804—31no—
52#1770—31no—
53#1762—32no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×C4×D3—22no—
58C22×C3×C8C2312yes—
59C3×C4×Q8—22no—
60C3×C4×Q8C2×C414yescomplemented
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#215—22no—
67C2×#215—22no—
68C2×#215C2×C414yes—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3D412yes—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79#893—24no—
80#893—42no—
81C4×D12—22no—
82C4×D12C2×C414yes—
83C3×C4⋊C8—22no—
84C3×#103—42no—
85C3×#95—21no—
86C3×#95—21no—
87#835C2×C414yescomplemented
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94C3⋊#95D411yescharacteristic
95C3⋊#95D411yescharacteristic
96#821—24no—
97#815—42no—
98#809C2×C414yes—
99#808C2×C414yescomplemented
100C4×C3⋊C8C2×C414yescomplemented
101C22×SD16—31no—
102C2×C4×Q8C2×D311yescharacteristic
103C2×C4×D4—32no—
104C2×C4×D4—31no—
105C4×SD16—38no—
106C2×#106—31no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×#103—31no—
109C2×#103C2×D312yes—
110C2×#102—31no—
111C2×#102—32no—
112C2×#97—32no—
113C2×C4×C8—31no—
114#391—32no—
115#389—32no—
116#387—62no—
117#385—62no—
118#383—61no—
119#382—31no—
120#382C2×D311yescharacteristic
121#380—61no—
122C4×C4⋊C4—31no—
123C4×C4⋊C4—32no—
124#376—61no—
125C2×#95—61no—
126#340—32no—
127#340—32no—
128#340—32no—
129#340C2×D312yes—
130#336—32no—
131C23×D3—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8C22×C412yes—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2411yescharacteristic
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3C2×D412yes—
147C22×Dic3C2×D412yes—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12C22×C412yes—
152C2×D12#4112yes—
153C2×D12#4112yes—
154C2×C4×D3—24no—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C4×D3—82no—
158C2×C4×D3—82no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4C22×C412yes—
167C3×C42—24no—
168C3×C42C22×C412yes—
169C3×C42#4112yes—
170#215—28no—
171(C2×D3)⋊C4—24no—
172(C2×D3)⋊C4—44no—
173C4⋊Dic3—21no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—42no—
177C4⋊Dic3C22×C412yes—
178Dic3⋊C4—82no—
179C4×Dic3—22no—
180C4×Dic3—22no—
181C4×Dic3—44no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8C22×C412yes—
187C2×C3⋊C8C22×C412yes—
188C22×Q8C22×D311yescharacteristic
189C22×D4—31no—
190C23×C4—62no—
191C23×C4—61no—
192C2×SD16—34no—
193C2×SD16—62no—
194C2×SD16—62no—
195C22×C8—31no—
196C22×C8—31no—
197C22×C8C22×D312yes—
198C4×Q8—22no—
199C4×Q8C4×D314yescomplemented
200C4×D4—34no—
201C4×D4—64no—
202C4×D4—62no—
203C4×D4—122no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4C22×D312yes—
210C2×C4⋊C4C22×D311yescharacteristic
211C2×C22⋊C4—61no—
212C2×C22⋊C4—62no—
213C2×C42—21no—
214C2×C42—32no—
215C2×C42—62no—
216C2×C42C22×D311yescharacteristic
217#106—34no—
218C4⋊C8—22no—
219#103—34no—
220#103—42no—
221#102—34no—
222#102—64no—
223#97—122no—
224C4×C8—34no—
225#95—21no—
226#95—21no—
227#95—31no—
228#95—31no—
229#95—62no—
230#95—62no—
231#95—62no—
232#95—62no—
233#95—61no—
234#95—61no—
235C23×C3C22×D411yescharacteristic
236C22×D3—22no—
237C22×D3—22no—
238C22×D3—22no—
239C22×D3—42no—
240C22×D3—42no—
241C3×Q8—22no—
242C3×Q8—22no—
243C3×Q8—41no—
244C3×Q8—41no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4C4×D412yes—
256C2×C3×C4C4×D412yes—
257C2×C3×C4C23×C411yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4H(5,𝔽2)11yescharacteristic
262C2×C3×C42−1+411yescharacteristic
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274D12—22no—
275D12—22no—
276D12—22no—
277D12#11714yes—
278C4×D3—84no—
279C4×D3—88no—
280C3×C8—42no—
281C3⋊C8—22no—
282C3⋊C8—22no—
283C24—61no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C2×C4×D312yes—
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—32no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—61no—
302C22×C4—64no—
303C22×C4—62no—
304C22×C4—124no—
305C22×C4—124no—
306C22×C4—242no—
307C22×C4—242no—
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4C23×D311yescharacteristic
311SD16—68no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—32no—
316C2×C8—32no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C4⋊C4—24no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—32no—
326C4⋊C4—61no—
327C4⋊C4—62no—
328C4⋊C4—62no—
329C4⋊C4—122no—
330C4⋊C4—242no—
331C4⋊C4C2×C4×D312yes—
332C22⋊C4—64no—
333C22⋊C4—124no—
334C42—24no—
335C42—62no—
336C42—62no—
337C42—124no—
338C42C2×C4×D312yes—
339C42#23712yes—
340C22×C3C2×C4×D411yescharacteristic
341C22×C3#54611yescharacteristic
342C22×C3#54811yescharacteristic
343C22×C3#57512yes—
344C22×C3#57712yes—
345C2×D3—24no—
346C2×D3—44no—
347C2×D3—42no—
348C2×D3—42no—
349C2×D3—44no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—82no—
355C3×C4C2×#11711yescharacteristic
356C3×C4#52011yescharacteristic
357C3×C4#55211yescharacteristic
358C3×C4#55311yescharacteristic
359Dic3—42no—
360Dic3—42no—
361Dic3—42no—
362Dic3—82no—
363C23—62no—
364C23—62no—
365C23—62no—
366C23—122no—
367C23—122no—
368C23C2×D3×D411yescharacteristic
369Q8—22no—
370Q8—22no—
371Q8—41no—
372Q8—41no—
373D4—34no—
374D4—62no—
375D4—62no—
376D4—62no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—244no—
399C2×C4—248no—
400C2×C4#67⋊C412yes—
401C2×C4#67⋊C412yes—
402C2×C4C22×C4×D311yescharacteristic
403C2×C4C2×#23711yescharacteristic
404C2×C4#100411yescharacteristic
405C2×C4#100711yescharacteristic
406C2×C4D3×#4111yescharacteristic
407C2×C4D3×#4111yescharacteristic
408C8—42no—
409C8—62no—
410C8—62no—
411C2×C3#235211yescharacteristic
412C2×C3#294311yescharacteristic
413C2×C3#294911yescharacteristic
414C2×C3#332312yes—
415C2×C3#337612yes—
416D3—44no—
417C22—64no—
418C22—122no—
419C22—124no—
420C22—122no—
421C22—124no—
422C22C2×#67⋊C411yescharacteristic
423C22#561211yescharacteristic
424C22#563311yescharacteristic
425C22#576812yes—
426C22#577812yes—
427C4—22no—
428C4—22no—
429C4—42no—
430C4—42no—
431C4—82no—
432C4—122no—
433C4—122no—
434C4—122no—
435C4—242no—
436C4#550411yescharacteristic
437C4D3×#11711yescharacteristic
438C4#561411yescharacteristic
439C4#563911yescharacteristic
440C3#2276011yescharacteristic, complemented, Sylow
441C2—124no—
442C2#7401211yescharacteristic
443C2#7881911yescharacteristic
444C2#7894711yescharacteristic
445C2#8062112yes—
446C2#8126512yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1805D3
hypocenterC3#22760
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. #82463C2semidirect
4. C3×#1805C2semidirect
6. #70466C2semidirect
8. #70256C2semidirect
9. #69547C2semidirect
10. #68788C2semidirect
24. C4×#215C4semidirect
50. #1805D3semidirect
60. C3×C4×Q8C2×C4semidirect
87. #835C2×C4semidirect
99. #808C2×C4semidirect
100. C4×C3⋊C8C2×C4semidirect
199. C4×Q8C4×D3semidirect
440. C3#22760semidirect

Maximal subgroups

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

subgroupindex
#824632
C2×C4×#2152
#742282
C3×#18052
#704802
#704662
#702762
#702562
#695472
#687882
#687822
#227603