Finite Group Databasebeta

← #10799592

Subgroups — #10799592

Order 768.

722 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82514C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3#78520C211yesmaximal, characteristic, complemented
4C3×#1759C211yesmaximal, characteristic, complemented
5#70373C212yesmaximal
6#70372C212yesmaximal, complemented
7C4×#822C211yesmaximal, characteristic, complemented
8#69434C211yesmaximal, characteristic, complemented
9#69430C211yesmaximal, characteristic, complemented
10#68747C212yesmaximal, complemented
11#68746C212yesmaximal, complemented
12#21294—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×C3×#104C2212yes—
17C3×C4×M16C2211yescharacteristic
18C3×#376C2211yescharacteristic
19#5234C2212yes—
20#5233C2211yescharacteristic
21#5129C2212yes—
22C2×#822C2212yes—
23#5119C2212yes—
24C4×#216C412yescomplemented
25C4×#215C412yescomplemented
26C4×#214C412yescomplemented
27C4×#213C412yescomplemented
28#5002C2211yescharacteristic
29#5000—22no—
30#4999C2212yes—
31#4998C2211yescharacteristic
32C2×#805C2212yes—
33#4926—22no—
34C4×#208C2211yescharacteristic
35C2×#803C2212yes—
36C2×#803C2211yescharacteristic
37C2×C4×C3⋊C8C2212yes—
38C2×C4×C3⋊C8C2211yescharacteristic
39#4705—22no—
40#4690—22no—
41C3×#339C2212yes—
42#4558C2212yes—
43#4523C2212yes—
44C4×#135—31no—
45#2923—31no—
46#2923—31no—
47#1974—32no—
48#1945—32no—
49#1938—32no—
50#1938—32no—
51C4×#108—31no—
52#1759—31no—
53#1759D311yescharacteristic, complemented
54#1726—31no—
55C2×C3×#41C2311yescharacteristic
56C2×#235C2311yescharacteristic
57C2×C3×M16C2312yes—
58C3×C4×Q8C2×C412yes—
59C3×C4×D4—22no—
60C3×C4×D4C2×C412yes—
61C3×#117—21no—
62C3×#117C2311yescharacteristic
63C2×C3×C42—21no—
64C2×C3×C42C2312yes—
65C2×C3×C42C2311yescharacteristic
66#950—24no—
67C2×#216C2×C411yescharacteristic
68C2×#215C2×C411yescharacteristic
69C2×#214C2×C411yescharacteristic
70C2×#213C2×C411yescharacteristic
71C4×#67—22no—
72#926C2311yescharacteristic
73C2×#208C2312yes—
74C22×C3⋊C8C2311yescharacteristic
75C22×C3⋊C8C2312yes—
76C22×C3⋊C8C2311yescharacteristic
77C4×D12C2×C412yes—
78#872—21no—
79C42×D3—21no—
80C4×Dic6C2×C412yes—
81C3×#104—24no—
82C3×#97—21no—
83C3×C4×C8—21no—
84C3×#95—22no—
85#835C2×C412yes—
86#832C2×C412yes—
87#830—24no—
88#822—24no—
89(C2×D3)⋊C8—24no—
90Dic3⋊C8—24no—
91#810C2×C412yes—
92#809C2×C412yes—
93#808C2×C412yes—
94#807C2×C412yes—
95#805—24no—
96C4⋊(C3⋊C8)—24no—
97#803—21no—
98#803—24no—
99#803D412yes—
100#803D412yes—
101C4×C3⋊C8—21no—
102C4×C3⋊C8—24no—
103C4×C3⋊C8C2×C412yes—
104C4×C3⋊C8C2×C412yes—
105C2×#135—31no—
106C4×#41—31no—
107C4×#41C2×D311yescharacteristic
108C4×Q16—32no—
109C4×SD16—32no—
110C4×SD16—32no—
111C4×D8—32no—
112#433—62no—
113#433—62no—
114#432—62no—
115#432—62no—
116C2×#108—32no—
117#427—31no—
118#424—32no—
119#424—32no—
120C2×#104—32no—
121C2×#104C2×D312yes—
122#416—31no—
123#416—31no—
124#408—32no—
125#408—32no—
126C4×M16—31no—
127C4×M16C2×D311yescharacteristic
128C2×#97—31no—
129C2×#97—32no—
130C2×C4×C8—31no—
131C2×C4×C8—32no—
132#376C2×D311yescharacteristic
133#341—32no—
134#339—32no—
135#339C2×D312yes—
136C3×#41—22no—
137C3×#41—22no—
138C2×C3×Q8C22×C411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4C22×C411yescharacteristic
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4#4112yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—21no—
147#235—22no—
148#235—22no—
149C2×D12C22×C411yescharacteristic
150C2×C4×D3—21no—
151C2×C4×D3—22no—
152C2×Dic6C22×C411yescharacteristic
153C3×M16—22no—
154C3×M16—22no—
155C2×C3×C8—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4C22×C412yes—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—22no—
160C3×C42—22no—
161C3×C42—21no—
162C3×C42—22no—
163C3×C42C22×C412yes—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4114yes—
167#217—21no—
168#216—22no—
169#215—22no—
170#214—22no—
171#213—22no—
172(C2×D3)⋊C4—22no—
173C4⋊Dic3C22×C412yes—
174Dic3⋊C4—22no—
175C4×Dic3—22no—
176C4×Dic3—21no—
177#208—22no—
178#208—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8C22×C412yes—
184C2×C3⋊C8C22×C411yescharacteristic
185C2×C3⋊C8C22×C411yescharacteristic
186C2×C3⋊C8C2×D414yes—
187C2×C3⋊C8#4114yes—
188C2×#41—31no—
189C2×#41C22×D311yescharacteristic
190#135—64no—
191C2×Q16—31no—
192C2×SD16—31no—
193C2×SD16—31no—
194C2×D8—31no—
195C2×M16—32no—
196C2×M16C22×D312yes—
197C22×C8—31no—
198C22×C8—31no—
199C22×C8—32no—
200C4×Q8—32no—
201C4×Q8C4×D312yes—
202C4×D4—22no—
203C4×D4—32no—
204C4×D4—62no—
205C4×D4C4×D312yes—
206#117—21no—
207#117—31no—
208#117—61no—
209#117C22×D311yescharacteristic
210C2×C42—21no—
211C2×C42—61no—
212C2×C42C22×D312yes—
213C2×C42C22×D311yescharacteristic
214#108—64no—
215#107—32no—
216#106—32no—
217C4⋊C8—64no—
218C4⋊C8—64no—
219#104—24no—
220#104—64no—
221#103—32no—
222#103—32no—
223#102—32no—
224#102—32no—
225C22⋊C8—64no—
226C22⋊C8—64no—
227#97—21no—
228#97—32no—
229#97—32no—
230#97—64no—
231#97—61no—
232C4×C8—21no—
233C4×C8—32no—
234C4×C8—32no—
235C4×C8—64no—
236C4×C8—61no—
237#95—22no—
238C23×C3—21no—
239C23×C3C2×#4111yescharacteristic
240C22×D3—21no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—42no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—42no—
250C2×C3×C4—44no—
251C2×C3×C4—42no—
252C2×C3×C4—82no—
253C2×C3×C4C4×D412yes—
254C2×C3×C4C4×D412yes—
255C2×C3×C4C23×C411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C4C2×#4112yes—
259#67—42no—
260C2×Dic3—21no—
261C2×Dic3—22no—
262D12—21no—
263D12—21no—
264C4×D3—42no—
265C4×D3—82no—
266Dic6—21no—
267Dic6—21no—
268C3×C8—42no—
269C3⋊C8—22no—
270C3⋊C8—24no—
271C3⋊C8—42no—
272C3⋊C8#11712yes—
273C3⋊C8#11712yes—
274#41—22no—
275#41—22no—
276#41—62no—
277#41—62no—
278C2×Q8—31no—
279C2×Q8C2×C4×D311yescharacteristic
280C2×D4—21no—
281C2×D4—31no—
282C2×D4—61no—
283C2×D4C2×C4×D311yescharacteristic
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4#23712yes—
290C22×C4C23×D311yescharacteristic
291Q16—62no—
292SD16—62no—
293SD16—62no—
294D8—62no—
295M16—22no—
296M16—22no—
297M16—62no—
298M16—62no—
299C2×C8—22no—
300C2×C8—31no—
301C2×C8—31no—
302C2×C8—34no—
303C2×C8—32no—
304C2×C8—34no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—62no—
309C4⋊C4—22no—
310C4⋊C4—32no—
311C4⋊C4—62no—
312C4⋊C4C2×C4×D312yes—
313C22⋊C4—21no—
314C22⋊C4—22no—
315C22⋊C4—61no—
316C22⋊C4—62no—
317C42—22no—
318C42—21no—
319C42—22no—
320C42—62no—
321C42—61no—
322C42C2×C4×D312yes—
323C42#23514yes—
324C42D3×D411yescharacteristic
325C42D3×D411yescharacteristic
326C22×C3—22no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3C2×C4×D411yescharacteristic
330C22×C3#52912yes—
331C2×D3—42no—
332C2×D3—41no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—44no—
336C3×C4—42no—
337C3×C4—41no—
338C3×C4C2×#11711yescharacteristic
339C3×C4C4×#4111yescharacteristic
340C3×C4#53212yes—
341Dic3—42no—
342Dic3—41no—
343C23—21no—
344C23—61no—
345C23C2×#23711yescharacteristic
346Q8—21no—
347Q8—21no—
348Q8—61no—
349Q8—61no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—61no—
354D4—61no—
355D4—122no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—62no—
363C2×C4—61no—
364C2×C4—82no—
365C2×C4—122no—
366C2×C4—242no—
367C2×C4#67⋊C412yes—
368C2×C4#67⋊C412yes—
369C2×C4C22×C4×D311yescharacteristic
370C2×C4C2×#23512yes—
371C2×C4C2×D3×D411yescharacteristic
372C2×C4D3×#4112yes—
373C8—32no—
374C8—32no—
375C8—42no—
376C8—62no—
377C8—64no—
378C8—122no—
379C2×C3—22no—
380C2×C3—81no—
381C2×C3#229311yescharacteristic
382C2×C3#295712yes—
383D3—81no—
384C22—22no—
385C22—42no—
386C22—41no—
387C22—122no—
388C22—121no—
389C22C2×#67⋊C411yescharacteristic
390C22#553812yes—
391C4—22no—
392C4—22no—
393C4—44no—
394C4—42no—
395C4—41no—
396C4—122no—
397C4—121no—
398C4C4×#23511yescharacteristic
399C4D3×#11711yescharacteristic
400C4#554812yes—
401C3#2129411yescharacteristic, complemented, Sylow
402C2—22no—
403C2—81no—
404C2—241no—
405C2#7343911yescharacteristic
406C2#7899012yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1759C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1759D3
hypocenterC3#21294
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
1. #82514C2semidirect
3. #78520C2semidirect
4. C3×#1759C2semidirect
6. #70372C2semidirect
7. C4×#822C2semidirect
8. #69434C2semidirect
9. #69430C2semidirect
10. #68747C2semidirect
11. #68746C2semidirect
24. C4×#216C4semidirect
25. C4×#215C4semidirect
26. C4×#214C4semidirect
27. C4×#213C4semidirect
53. #1759D3semidirect
401. C3#21294semidirect

Maximal subgroups

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

subgroupindex
#825142
C4×#9502
#785202
C3×#17592
#703732
#703722
C4×#8222
#694342
#694302
#687472
#687462
#212943