Finite Group Databasebeta

← #10804465

Subgroups — #10804465

Order 768.

1258 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic
2#79038C211yesmaximal, characteristic
3#78517C211yesmaximal, characteristic
4C3×#1966C211yesmaximal, characteristic, complemented
5#71789C211yesmaximal, characteristic, complemented
6#70522C211yesmaximal, characteristic, complemented
7#70113C214yesmaximal
8#70098C214yesmaximal, complemented
9#69877C211yesmaximal, characteristic, complemented
10#21799—31nomaximal, Sylow
11C22×D3×Q8C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×#944C2212yes—
14D3×C4⋊Q8—22no—
15D3×C4⋊Q8—22no—
16D3×C4⋊Q8—22no—
17D3×C4⋊Q8—22no—
18C2×D3×C4⋊C4C414yes—
19C2×#888C414yes—
20C2×C42×D3C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×#423C2211yescharacteristic
23C2×C3×#101C2214yes—
24C3×C4×M16C2211yescharacteristic
25C2×#836C2214yes—
26C2×#824C2214yes—
27C2×#824C2214yes—
28#5126C2212yes—
29#5126C2212yes—
30#5118C2212yes—
31Dic3×M16C2212yes—
32#4924C2211yescharacteristic
33C4×#208C2211yescharacteristic
34#4835—22no—
35#4833—22no—
36#4705—22no—
37#4685—22no—
38C22×C4⋊Q8—31no—
39#2920—31no—
40#2920—31no—
41#1966—31no—
42#1966—34no—
43#1966—31no—
44#1966—31no—
45#1966D311yescharacteristic, complemented
46#1784—34no—
47C22×C3×Q8C2312yes—
48C2×D3×Q8—24no—
49C2×D3×Q8—24no—
50C2×D3×Q8—44no—
51C2×D3×Q8—44no—
52C22×C4×D3C2×C412yes—
53C22×C4×D3C2×C414yes—
54C22×C4×D3C2311yescharacteristic
55C22×Dic6C2×C414yes—
56C22×Dic6C2312yes—
57C2×C3×M16C2312yes—
58C2×C3×M16C2312yes—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—22no—
62C2×C3×C4⋊C4C2×C414yes—
63C2×C3×C42C2311yescharacteristic
64#944—22no—
65#944—24no—
66#944—22no—
67C2×C4⋊Dic3C2×C414yes—
68C2×Dic3⋊C4C2×C418yes—
69C2×C4×Dic3C2×C412yes—
70C2×C4×Dic3C2311yescharacteristic
71C2×#208C2312yes—
72C2×#208C2312yes—
73D3×C4⋊C4—24no—
74D3×C4⋊C4—24no—
75D3×C4⋊C4—24no—
76D3×C4⋊C4—24no—
77#888—24no—
78#888—24no—
79#888—24no—
80#888—24no—
81C42×D3D412yes—
82C42×D3D412yes—
83C42×D3D412yes—
84C42×D3D412yes—
85#869—21no—
86#869—21no—
87#869—22no—
88C3×C4⋊C8—22no—
89C3×#101—44no—
90C3×#97—21no—
91C3×C4×C8—21no—
92#836—44no—
93#824—44no—
94#824—44no—
95(C2×D3)⋊C8—24no—
96(C2×D3)⋊C8—24no—
97#815—22no—
98Dic3⋊C8—24no—
99C8×Dic3—22no—
100C4⋊(C3⋊C8)—22no—
101#803—21no—
102C4×C3⋊C8—21no—
103C23×Q8—32no—
104C2×C4⋊Q8—32no—
105C2×C4⋊Q8—34no—
106C2×C4⋊Q8—31no—
107C2×C4⋊Q8—62no—
108C2×C4⋊Q8—62no—
109C2×C4⋊Q8—62no—
110C2×C4⋊Q8—62no—
111C2×C4⋊Q8C2×D311yescharacteristic
112C22×C4⋊C4—34no—
113C22×C42—31no—
114#432—62no—
115#432—62no—
116#431—62no—
117#431—62no—
118#423—32no—
119#423—31no—
120#423C2×D311yescharacteristic
121C2×#101—34no—
122C2×#101—34no—
123C2×#101—34no—
124C2×#101C2×D314yes—
125#407—32no—
126#407—32no—
127C4×M16—32no—
128C4×M16—31no—
129C4×M16C2×D311yescharacteristic
130C23×D3C22×C411yescharacteristic
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—24no—
135C22×C3×C4C22×C414yes—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C2411yescharacteristic
138C22×Dic3C22×C414yes—
139C22×Dic3C22×C411yescharacteristic
140C22×Dic3C22×C412yes—
141D3×Q8—48no—
142D3×Q8—48no—
143D3×Q8—416no—
144C2×C4×D3—22no—
145C2×C4×D3—22no—
146C2×C4×D3—28no—
147C2×C4×D3—28no—
148C2×C4×D3—48no—
149C2×C4×D3C22⋊C412yes—
150C2×C4×D3C22⋊C412yes—
151C2×C4×D3C22⋊C412yes—
152C2×C4×D3C22⋊C412yes—
153C2×C4×D3C2×D412yes—
154C2×C4×D3C2×D412yes—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6—24no—
158C2×Dic6—24no—
159C2×Dic6—28no—
160C2×Dic6—48no—
161C3×M16—22no—
162C3×M16—24no—
163C3×M16—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D412yes—
171C4⋊Dic3—24no—
172C4⋊Dic3—24no—
173Dic3⋊C4—28no—
174Dic3⋊C4—28no—
175C4×Dic3C22⋊C412yes—
176C4×Dic3C22⋊C412yes—
177C4×Dic3C22⋊C412yes—
178C4×Dic3C22⋊C412yes—
179C4×Dic3C2×D411yescharacteristic
180C4×Dic3C2×D412yes—
181C4×Dic3C2×D411yescharacteristic
182#208—22no—
183#208—24no—
184#208—22no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C22×Q8—34no—
188C22×Q8—32no—
189C22×Q8—64no—
190C22×Q8—64no—
191C22×Q8—124no—
192C22×Q8—124no—
193C22×Q8C22×D312yes—
194C23×C4—32no—
195C23×C4—31no—
196C23×C4—34no—
197C2×M16—32no—
198C2×M16—32no—
199C2×M16C22×D312yes—
200C2×M16C22×D312yes—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—22no—
204C4⋊Q8—64no—
205C4⋊Q8—64no—
206C4⋊Q8—64no—
207C4⋊Q8—64no—
208C4⋊Q8—62no—
209C4⋊Q8—62no—
210C4⋊Q8—64no—
211C4⋊Q8—61no—
212C4⋊Q8—61no—
213C4⋊Q8—62no—
214C2×C4⋊C4—38no—
215C2×C4⋊C4—34no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4C4×D314yes—
221C2×C42—32no—
222C2×C42—31no—
223C2×C42—32no—
224C2×C42—32no—
225C2×C42—32no—
226C2×C42—32no—
227C2×C42C22×D311yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—64no—
230C4⋊C8—62no—
231#101—44no—
232#101—124no—
233#101—124no—
234#101—124no—
235C22⋊C8—64no—
236C22⋊C8—64no—
237#97—21no—
238#97—62no—
239#97—61no—
240C4×C8—21no—
241C4×C8—62no—
242C4×C8—61no—
243C23×C3C23×C411yescharacteristic
244C22×D3—22no—
245C22×D3—24no—
246C22×D3C2×C22⋊C412yes—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×Q8—44no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—28no—
253C2×C3×C4C2×C22⋊C412yes—
254C2×C3×C4C2×C22⋊C412yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4H(5,𝔽2)12yes—
258C2×C3×C4H(5,𝔽2)12yes—
259C2×Dic3—22no—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—28no—
263C2×Dic3C2×C22⋊C412yes—
264C2×Dic3C2×C22⋊C412yes—
265C2×Dic3C2×C22⋊C412yes—
266C4×D3—24no—
267C4×D3—22no—
268C4×D3—22no—
269C4×D3—24no—
270C4×D3—24no—
271C4×D3—24no—
272C4×D3—416no—
273C4×D3#10118yescomplemented
274Dic6—416no—
275Dic6—42no—
276Dic6—42no—
277Dic6—44no—
278C3×C8—42no—
279C3×C8—42no—
280C3⋊C8—42no—
281C3⋊C8—42no—
282C24—31no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—68no—
288C2×Q8—62no—
289C2×Q8—62no—
290C2×Q8—64no—
291C2×Q8—64no—
292C2×Q8—128no—
293C2×Q8—128no—
294C2×Q8—128no—
295C2×Q8—1216no—
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—31no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—34no—
305C22×C4—68no—
306C22×C4—68no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—128no—
310C22×C4C2×C4×D314yes—
311C22×C4C2×C4×D312yes—
312C22×C4C23×D311yescharacteristic
313M16—22no—
314M16—24no—
315M16—22no—
316M16—62no—
317M16—64no—
318M16—62no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—62no—
322C2×C8—62no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—68no—
326C4⋊C4—68no—
327C4⋊C4—64no—
328C4⋊C4—64no—
329C42—32no—
330C42—32no—
331C42—32no—
332C42—32no—
333C42—31no—
334C42—31no—
335C42—32no—
336C42D3×D411yescharacteristic
337C42D3×D411yescharacteristic
338C42D3×D412yes—
339C22×C3—22no—
340C22×C3C22×C22⋊C411yescharacteristic
341C22×C3#51812yes—
342C2×D3—28no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3C2×#10114yes—
346C3×C4—22no—
347C3×C4—44no—
348C3×C4C2×#10114yes—
349C3×C4#53511yescharacteristic
350C3×C4#53511yescharacteristic
351C3×C4#53512yes—
352Dic3—22no—
353Dic3—22no—
354Dic3—44no—
355Dic3C2×#10114yes—
356C23—32no—
357C23—64no—
358C23—62no—
359C23C22×C4×D311yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—44no—
363Q8—1216no—
364Q8—122no—
365Q8—122no—
366Q8—124no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—28no—
370C2×C4—32no—
371C2×C4—38no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—64no—
375C2×C4—64no—
376C2×C4—64no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—64no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—68no—
384C2×C4—1216no—
385C2×C4D3×C22⋊C412yes—
386C2×C4D3×C22⋊C412yes—
387C2×C4C2×D3×D411yescharacteristic
388C2×C4C2×D3×D411yescharacteristic
389C2×C4#100412yes—
390C2×C4#100912yes—
391C8—42no—
392C8—42no—
393C8—122no—
394C8—122no—
395C2×C3—22no—
396C2×C3#230911yescharacteristic
397C2×C3C22×#10112yes—
398D3—24no—
399C22—22no—
400C22—34no—
401C22—68no—
402C22—62no—
403C22—62no—
404C22C2×D3×C22⋊C411yescharacteristic
405C22#554512yes—
406C4—22no—
407C4—34no—
408C4—44no—
409C4—62no—
410C4—62no—
411C4—124no—
412C4D3×#10114yes—
413C4#560712yes—
414C4#567411yescharacteristic
415C4#571511yescharacteristic
416C3#2179911yescharacteristic, complemented, Sylow
417C2—22no—
418C2—64no—
419C2#7642811yescharacteristic
420C2C2×D3×#10112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1966C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22×C4
cosocleC22×C3×C4C24
hypercenter#1966D3
hypocenterC3#21799
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
4. C3×#1966C2semidirect
5. #71789C2semidirect
6. #70522C2semidirect
8. #70098C2semidirect
9. #69877C2semidirect
45. #1966D3semidirect
273. C4×D3#101semidirect
416. C3#21799semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
#790382
#785172
C3×#19662
#717892
#705222
#701132
#700982
#698772
#217993