Finite Group Databasebeta

← #10691300

Subgroups — #10691300

Order 768.

536 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3106C211yesmaximal, characteristic
2C4×#948C211yesmaximal, characteristic, complemented
3#71802C211yesmaximal, characteristic, complemented
4#71797C211yesmaximal, characteristic, complemented
5#71641C211yesmaximal, characteristic, complemented
6#67743C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#9211—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C2×#948—21no—
12C3×#482—21no—
13C3×#477C411yescharacteristic
14C3×#476C411yescharacteristic
15C3×#460C411yescharacteristic
16C3×#459C411yescharacteristic
17C3×#449—21no—
18C3×C4×SD16—21no—
19C3×C4×D8—21no—
20C3×#431C2211yescharacteristic
21#4997—21no—
22#4997—22no—
23#4924C2211yescharacteristic
24C4×#208—21no—
25C4×#208C2211yescharacteristic
26C4×#208C2211yescharacteristic
27C2×C4×C3⋊C8—21no—
28C2×C4×C3⋊C8C2211yescharacteristic
29#4540—22no—
30#4535—21no—
31D4⋊(C3⋊C8)—21no—
32D4⋊(C3⋊C8)—21no—
33D4⋊(C3⋊C8)—21no—
34#3106D311yescharacteristic
35C4×#131—31no—
36#1525—31no—
37#1492—31no—
38#1489—31no—
39#1299—31no—
40#1278—31no—
41C2×C3×#41—21no—
42C2×C3×#41D411yescharacteristic
43C2×C3×SD16—21no—
44C2×C3×D8—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8C2×C411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4C2×C411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C3×#117D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62#948—24no—
63#948—24no—
64C2×#208—22no—
65C2×#208—21no—
66C2×#208—21no—
67C2×#208—21no—
68C22×C3⋊C8—22no—
69C22×C3⋊C8—21no—
70C3×#107—21no—
71C3×#106—21no—
72C3×C4⋊C8C2×C411yescharacteristic
73C3×C4⋊C8C2×C411yescharacteristic
74C3×#103—21no—
75C3×#102—21no—
76C3×#102—21no—
77C3×#102—21no—
78C3×C22⋊C8—21no—
79C3×C4×C8—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—22no—
83#803—21no—
84#803D412yes—
85C4×C3⋊C8—22no—
86C4×C3⋊C8—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D411yescharacteristic
90C4×C3⋊C8D411yescharacteristic
91C2×#131—61no—
92#532C2×D311yescharacteristic
93C4×#41C2×D311yescharacteristic
94#482—21no—
95#477Dic311yescharacteristic
96#476Dic311yescharacteristic
97#460Dic311yescharacteristic
98#459Dic311yescharacteristic
99#449—21no—
100C4×SD16—21no—
101C4×D8—21no—
102#431C2×D311yescharacteristic
103#423—31no—
104#405—62no—
105#405—61no—
106C4×M16—31no—
107C4×M16—31no—
108C4×M16—61no—
109C2×C4×C8—31no—
110C2×C4×C8—61no—
111#329—62no—
112Q8⋊C8—61no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—61no—
116C3×#41—22no—
117C3×#41—22no—
118C3×#41—81no—
119C3×#41—81no—
120C2×C3×Q8—21no—
121C2×C3×Q8C22⋊C411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C2×C3×D4C22⋊C411yescharacteristic
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C3×SD16—42no—
136C3×D8—42no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#208—28no—
158#208—22no—
159#208—22no—
160#208—22no—
161#208—41no—
162#208—81no—
163C2×C3⋊C8—28no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×#41—21no—
172C2×#41#6711yescharacteristic
173C2×SD16—21no—
174C2×D8—21no—
175#131—64no—
176#131—64no—
177C2×M16—62no—
178C2×M16—61no—
179C2×M16—61no—
180C2×M16—61no—
181C22×C8—62no—
182C22×C8—61no—
183C4⋊Q8C2×Dic311yescharacteristic
184#127C2×Dic311yescharacteristic
185#124—21no—
186#121—21no—
187#121—21no—
188C4×Q8C2×Dic311yescharacteristic
189C4×Q8#6711yescharacteristic
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4C2×Dic311yescharacteristic
195C4×D4#6711yescharacteristic
196#117—21no—
197#117#6711yescharacteristic
198C2×C42—21no—
199C2×C42C22×D311yescharacteristic
200#107—21no—
201#106—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8C2×Dic311yescharacteristic
205C4⋊C8C2×Dic311yescharacteristic
206#103—21no—
207#102—21no—
208#102—21no—
209#102—21no—
210C22⋊C8—21no—
211#97—32no—
212#97—62no—
213#97—61no—
214C4×C8—21no—
215C4×C8—31no—
216C4×C8—31no—
217C4×C8—62no—
218C4×C8—61no—
219C4×C8—61no—
220C4×C8—61no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3#12311yescharacteristic
224C3×Q8—21no—
225C3×Q8—81no—
226C3×Q8#10412yes—
227C3×D4—24no—
228C3×D4—21no—
229C3×D4—41no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—81no—
234C3×D4—82no—
235C3×D4#10412yes—
236C2×C3×C4—24no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4C2×C22⋊C411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4C4×D411yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C3×C8—41no—
260C3×C8—41no—
261C3⋊C8—22no—
262C3⋊C8—24no—
263C3⋊C8—22no—
264C3⋊C8—42no—
265C3⋊C8—81no—
266#41—22no—
267#41—22no—
268#41—81no—
269#41—81no—
270C2×Q8—21no—
271C2×Q8#21711yescharacteristic
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4#21711yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4#23711yescharacteristic
283C22×C4#23711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285SD16—42no—
286D8—42no—
287M16—68no—
288M16—62no—
289M16—62no—
290M16—62no—
291M16—121no—
292M16—241no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—68no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—62no—
301C2×C8—121no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4#21711yescharacteristic
308C4⋊C4#21711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42C22×Dic311yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#38611yescharacteristic
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—22no—
334C3×C4—21no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—81no—
338C3×C4C2×#10411yescharacteristic
339C3×C4C2×#10411yescharacteristic
340C3×C4#44311yescharacteristic
341C3×C4#44411yescharacteristic
342C3×C4#47111yescharacteristic
343C3×C4#47211yescharacteristic
344C23—21no—
345C23—41no—
346C23#93511yescharacteristic
347Q8—21no—
348Q8—81no—
349Q8#83712yes—
350D4—24no—
351D4—21no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—82no—
358D4#83712yes—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4D4×Dic311yescharacteristic
377C2×C4D4×Dic311yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#93911yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4C2×#21711yescharacteristic
382C8—41no—
383C8—41no—
384C8—62no—
385C8—62no—
386C8—64no—
387C8—122no—
388C8—241no—
389C2×C3—41no—
390C2×C3—42no—
391C2×C3—161no—
392C2×C3#187611yescharacteristic
393C2×C3#189611yescharacteristic
394C2×C3#190711yescharacteristic
395C22—21no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22#522011yescharacteristic
405C4—21no—
406C4—22no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—81no—
412C4#519611yescharacteristic
413C4#519711yescharacteristic
414C4#519911yescharacteristic
415C4#520011yescharacteristic
416C4C2×#83711yescharacteristic
417C4C2×#83711yescharacteristic
418C3#921111yescharacteristic, complemented, Sylow
419C2—41no—
420C2—42no—
421C2—161no—
422C2#7171111yescharacteristic
423C2#7208711yescharacteristic
424C2#7209111yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3106C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3106D3
hypocenterC3#9211
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
2. C4×#948C2semidirect
3. #71802C2semidirect
4. #71797C2semidirect
5. #71641C2semidirect
6. #67743C2semidirect
7. #67739C2semidirect
418. C3#9211semidirect

Maximal subgroups

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

subgroupindex
C3×#31062
C4×#9482
#718022
#717972
#716412
#677432
#677392
#92113