Finite Group Databasebeta

← #10814898

Subgroups — #10814898

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79000C212yesmaximal, complemented
2#78494C211yesmaximal, characteristic
3C3×#1771C211yesmaximal, characteristic, complemented
4#69845C211yesmaximal, characteristic
5#69842C211yesmaximal, characteristic, complemented
6#69829C211yesmaximal, characteristic, complemented
7#69795C211yesmaximal, characteristic
8#69779C211yesmaximal, characteristic, complemented
9#69494C212yesmaximal, complemented
10#68646C212yesmaximal, complemented
11#68646C212yesmaximal, complemented
12#22804—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14C4×#235C2212yes—
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×#97C2211yescharacteristic
17C2×C3×#97C2211yescharacteristic
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2211yescharacteristic
20#5113C2212yes—
21#5110C2212yes—
22#5110C2212yes—
23C2×#819C2211yescharacteristic
24C2×#819C2211yescharacteristic
25C2×#819C2212yes—
26#5106C2211yescharacteristic
27#5101C2212yes—
28#5101C2212yes—
29C2×#805C2212yes—
30C2×#805C2212yes—
31#4924C2212yes—
32#4840—22no—
33#4840—22no—
34#4835—22no—
35#4833—22no—
36#4716C412yescomplemented
37#4711C412yescomplemented
38#4710C412yescomplemented
39#4710C412yescomplemented
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42#4523C2212yes—
43#4523C2212yes—
44#2943—31no—
45#2925—32no—
46#1975—31no—
47#1974—31no—
48#1929—32no—
49#1929—32no—
50#1849—31no—
51#1806—32no—
52#1773—31no—
53#1772—31no—
54#1771D311yescharacteristic, complemented
55C2×#235C2312yes—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C2×C3×C4⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C4×#67—22no—
65C4×#67—22no—
66#926C2312yes—
67C2×#208C2312yes—
68C2×#208C2312yes—
69C2×Dic12C2×C411yescharacteristic
70#904—22no—
71#904—22no—
72C2×D24C2×C411yescharacteristic
73C2×#204C2×C412yes—
74C4×D12C2×C412yes—
75C4×D12C2×C412yes—
76#872—22no—
77C42×D3—22no—
78C4×Dic6C2×C412yes—
79C4×Dic6C2×C412yes—
80C3×C4⋊C8D412yes—
81C3×C4⋊C8D412yes—
82C3×#97—22no—
83C3×#97C2×C412yes—
84C3×#97C2×C412yes—
85C3×C4×C8—22no—
86C3×#95—21no—
87#821C2×C412yes—
88#821C2×C412yes—
89(C2×D3)⋊C8—22no—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—22no—
93#819—22no—
94#819—22no—
95#819—42no—
96#818C2×C411yescharacteristic
97#818C2×C411yescharacteristic
98#817C2×C412yes—
99#816C2×C412yes—
100#816C2×C412yes—
101#815—22no—
102#815—22no—
103C8×Dic3—22no—
104C8×Dic3—22no—
105#805—42no—
106#805—42no—
107C4⋊(C3⋊C8)—22no—
108C4⋊(C3⋊C8)—22no—
109C2×#135—31no—
110C4×#41—32no—
111#442—32no—
112#441—32no—
113#440—32no—
114#440—32no—
115#433—62no—
116#433—62no—
117#432—62no—
118#431—62no—
119C2×#108—31no—
120C2×#108—31no—
121C2×#108—32no—
122#427—31no—
123#423—32no—
124C2×C4⋊C8C2×D311yescharacteristic
125C2×#104—32no—
126C2×#104—32no—
127#416—32no—
128#408—32no—
129#408—32no—
130#405—32no—
131#405—32no—
132C2×#97C2×D311yescharacteristic
133C2×#97C2×D311yescharacteristic
134C2×C4×C8C2×D311yescharacteristic
135C4×C4⋊C4C2×D311yescharacteristic
136#339—32no—
137#339—32no—
138#336C2×D311yescharacteristic
139#336C2×D311yescharacteristic
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C2×#67—22no—
146#235—22no—
147#235—22no—
148#235—22no—
149#235—22no—
150C2×D12C22×C412yes—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×Dic6C22×C412yes—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8C22×C412yes—
162C2×C3×C8C22×C411yescharacteristic
163C2×C3×C8C22×C411yescharacteristic
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2×D412yes—
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8#4112yes—
168C2×C3×C8#4112yes—
169C3×C4⋊C4—42no—
170C3×C42—21no—
171C3×C42—21no—
172C3×C42—21no—
173C3×C42—21no—
174C3×C42C22×C412yes—
175C3×C42C2×D412yes—
176#217—22no—
177(C2×D3)⋊C4—22no—
178(C2×D3)⋊C4—22no—
179C4⋊Dic3C22×C412yes—
180C4⋊Dic3C22×C412yes—
181Dic3⋊C4—22no—
182Dic3⋊C4—22no—
183C4×Dic3—22no—
184C4×Dic3—22no—
185C4×Dic3—22no—
186#208—22no—
187#208—22no—
188#208—22no—
189#208—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192Dic12—22no—
193D24—22no—
194#204—24no—
195C2×#41—32no—
196#135—62no—
197#135—62no—
198C2×Q16—31no—
199C2×SD16—32no—
200C2×D8—31no—
201C2×M16—32no—
202C2×M16—32no—
203C22×C8C22×D311yescharacteristic
204C22×C8C22×D311yescharacteristic
205C22×C8C22×D311yescharacteristic
206C22×C8C22×D311yescharacteristic
207C4×Q8—32no—
208C4×Q8—32no—
209C4×D4—32no—
210C4×D4—32no—
211C4×D4—62no—
212C4×D4—62no—
213#117—32no—
214#117—62no—
215C2×C4⋊C4—21no—
216C2×C42—62no—
217C2×C42C22×D311yescharacteristic
218C2×C42C22×D311yescharacteristic
219C2×C42C22×D311yescharacteristic
220#108—62no—
221#108—62no—
222#108—122no—
223#107—31no—
224#107—31no—
225#106—32no—
226C4⋊C8—62no—
227C4⋊C8—62no—
228C4⋊C8D1212yes—
229C4⋊C8D1212yes—
230#104—122no—
231#104—122no—
232#103—32no—
233#103—32no—
234#102—32no—
235#102—32no—
236C22⋊C8—62no—
237C22⋊C8—62no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—22no—
241#97—62no—
242#97—62no—
243#97C4×D312yes—
244#97C4×D312yes—
245C4×C8—22no—
246C4×C8—62no—
247C4×C8—62no—
248#95—21no—
249C23×C3C2×#4111yescharacteristic
250C22×D3—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4C4×D412yes—
260C2×C3×C4C4×D412yes—
261C2×C3×C4C23×C411yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4H(5,𝔽2)11yescharacteristic
266C2×C3×C42−1+411yescharacteristic
267#67—42no—
268#67—42no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272D12—22no—
273D12—22no—
274C4×D3—44no—
275C4×D3—42no—
276C4×D3—42no—
277C4×D3—82no—
278Dic6—22no—
279Dic6—22no—
280C3×C8—22no—
281C3×C8—21no—
282C3×C8—21no—
283C3×C8—22no—
284C3×C8#11712yes—
285C3×C8#11712yes—
286C3⋊C8—42no—
287C3⋊C8—42no—
288#41—62no—
289#41—62no—
290#41—62no—
291#41—62no—
292C2×Q8—32no—
293C2×D4—32no—
294C2×D4—62no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4#23511yescharacteristic
301C22×C4#23511yescharacteristic
302C22×C4C23×D311yescharacteristic
303Q16—62no—
304SD16—64no—
305D8—62no—
306M16—62no—
307M16—62no—
308M16—62no—
309M16—62no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—62no—
317C2×C8—62no—
318C2×C8C2×C4×D312yes—
319C2×C8C2×C4×D311yescharacteristic
320C2×C8C2×C4×D311yescharacteristic
321C2×C8C2×D1211yescharacteristic
322C2×C8C2×D1212yes—
323C2×C8C2×D1211yescharacteristic
324C2×C8#23712yes—
325C2×C8#23912yes—
326C4⋊C4—32no—
327C4⋊C4—32no—
328C4⋊C4—42no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—62no—
339C42—62no—
340C42—62no—
341C42C2×C4×D312yes—
342C42C2×D1212yes—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3C2×C4×D411yescharacteristic
346C22×C3#55511yescharacteristic
347C22×C3#55611yescharacteristic
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—42no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—42no—
354C3×C4—81no—
355C3×C4C2×#11711yescharacteristic
356C3×C4#52011yescharacteristic
357C3×C4#54711yescharacteristic
358C3×C4#55011yescharacteristic
359Dic3—42no—
360Dic3—42no—
361Dic3—42no—
362C23—62no—
363C23C2×#23511yescharacteristic
364Q8—62no—
365Q8—62no—
366D4—62no—
367D4—62no—
368D4—122no—
369D4—122no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—124no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—242no—
385C2×C4C4×D1212yes—
386C2×C4C4×D1212yes—
387C2×C4C22×C4×D311yescharacteristic
388C2×C4C22×D1211yescharacteristic
389C2×C4C2×#23711yescharacteristic
390C2×C4C2×#23911yescharacteristic
391C2×C4#100911yescharacteristic
392C2×C4#101011yescharacteristic
393C8—22no—
394C8—21no—
395C8—21no—
396C8—22no—
397C8—122no—
398C8—122no—
399C8#89212yes—
400C8#89212yes—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3#235411yescharacteristic
404C2×C3#295511yescharacteristic
405C2×C3#295611yescharacteristic
406D3—82no—
407C22—21no—
408C22—21no—
409C22—122no—
410C22—122no—
411C22—122no—
412C22C2×C4×D1211yescharacteristic
413C22#556311yescharacteristic
414C22#558411yescharacteristic
415C4—22no—
416C4—22no—
417C4—42no—
418C4—81no—
419C4—122no—
420C4—122no—
421C4—122no—
422C4C2×#89211yescharacteristic
423C4#552311yescharacteristic
424C4#555511yescharacteristic
425C4#557711yescharacteristic
426C3#2280411yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—242no—
430C2#7363811yescharacteristic
431C2#7854111yescharacteristic
432C2#7857911yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1771C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1771D3
hypocenterC3#22804
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #1771

Complements

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

NH ≅ G/Ntype
1. #79000C2semidirect
3. C3×#1771C2semidirect
5. #69842C2semidirect
6. #69829C2semidirect
8. #69779C2semidirect
9. #69494C2semidirect
10. #68646C2semidirect
11. #68646C2semidirect
36. #4716C4semidirect
37. #4711C4semidirect
38. #4710C4semidirect
39. #4710C4semidirect
54. #1771D3semidirect
426. C3#22804semidirect

Maximal subgroups

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

subgroupindex
#790002
#784942
C3×#17712
#698452
#698422
#698292
#697952
#697792
#694942
#686462
#228043