Finite Group Databasebeta

← #10691272

Subgroups — #10691272

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3076C211yesmaximal, characteristic
2#41×C3⋊C8C211yesmaximal, characteristic
3#71835C211yesmaximal, characteristic, complemented
4#71826C211yesmaximal, characteristic, complemented
5#71646C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9207—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C3×#479—21no—
12C3×#478—21no—
13C3×#477C411yescharacteristic, complemented
14C3×#476C411yescharacteristic, complemented
15C3×#475C411yescharacteristic, complemented
16C3×#474C411yescharacteristic, complemented
17C3×#427—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#416—21no—
20C2×#837—21no—
21Q8×C3⋊C8—21no—
22D4×C3⋊C8—21no—
23D4×C3⋊C8—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C2×C4×C3⋊C8—21no—
29C2×C4×C3⋊C8C2211yescharacteristic
30#4538—22no—
31#4535—21no—
32#4535—21no—
33#4532—21no—
34#3076D311yescharacteristic
35C8×#41—31no—
36#1806—31no—
37#1503—31no—
38#1493—31no—
39#1301—31no—
40#1279—31no—
41C2×C3×#41D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8C2×C411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4C2×C411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C3×#117—21no—
57C3×#117—21no—
58C3×#117D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C22×C3⋊C8—22no—
65C22×C3⋊C8—21no—
66C3×#107—21no—
67C3×#107—21no—
68C3×#106—21no—
69C3×#106—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8C2×C411yescharacteristic
72C3×C4⋊C8C2×C411yescharacteristic
73C3×#103—21no—
74C3×#103—21no—
75C3×#102—21no—
76C3×#102—21no—
77#837—42no—
78C3⋊#95—21no—
79#830—22no—
80#830D412yes—
81C4⋊(C3⋊C8)—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4×C3⋊C8—22no—
86C4×C3⋊C8—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8D411yescharacteristic
89C4×C3⋊C8D411yescharacteristic
90#533C2×D311yescharacteristic
91C4×#41C2×D311yescharacteristic
92#479—21no—
93#478—21no—
94#477Dic311yescharacteristic, complemented
95#476Dic311yescharacteristic, complemented
96#475Dic311yescharacteristic, complemented
97#474Dic311yescharacteristic, complemented
98C8×Q8—61no—
99C8×D4—62no—
100C8×D4—61no—
101#431—31no—
102#431—61no—
103#427—21no—
104#423—31no—
105C2×C4⋊C8C2×D311yescharacteristic
106C2×#104—61no—
107#416—21no—
108C2×C4×C8—31no—
109C2×C4×C8—61no—
110C4×C4⋊C4—31no—
111#339—61no—
112#328—62no—
113Q8⋊C8—61no—
114Q8⋊C8—61no—
115C3×#41—21no—
116C3×#41—21no—
117C3×#41SD1612yes—
118C3×#41Q1612yes—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8C22⋊C411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4C22⋊C411yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—41no—
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
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×C42—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—41no—
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157C4⋊Dic3—22no—
158C4⋊Dic3—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—42no—
167C2×C3⋊C8—44no—
168C2×C3⋊C8—41no—
169C2×#41#6711yescharacteristic
170C2×M16—61no—
171C22×C8—21no—
172C22×C8—62no—
173C22×C8—61no—
174C4⋊Q8C2×Dic311yescharacteristic
175C4⋊Q8C2×Dic311yescharacteristic
176#125—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8C2×Dic311yescharacteristic
182C4×Q8#6711yescharacteristic
183C4×D4—21no—
184C4×D4—21no—
185C4×D4C2×Dic311yescharacteristic
186C4×D4#6711yescharacteristic
187#117—21no—
188#117—21no—
189#117#6711yescharacteristic
190C2×C4⋊C4—61no—
191C2×C42—21no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194#107—21no—
195#107—21no—
196#106—21no—
197#106—21no—
198C4⋊C8—21no—
199C4⋊C8—62no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8C2×Dic311yescharacteristic
204C4⋊C8C2×Dic311yescharacteristic
205#104—122no—
206#103—21no—
207#103—21no—
208#102—21no—
209#102—21no—
210C22⋊C8—32no—
211C22⋊C8—62no—
212C4×C8—31no—
213C4×C8—31no—
214C4×C8—62no—
215C4×C8—61no—
216C4×C8—61no—
217#95—61no—
218C23×C3—21no—
219C23×C3#12111yescharacteristic
220C3×Q8—21no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—81no—
224C3×Q8#10312yes—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—21no—
228C3×D4#10312yes—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4C2×C22⋊C411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248C2×C3×C4C2×SD1611yescharacteristic
249C2×C3×C4C2×Q1611yescharacteristic
250C2×C3×C4#13511yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×Dic3—42no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C3×C8—41no—
256C3×C8—41no—
257C3⋊C8—22no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C3⋊C8—81no—
261#41—21no—
262#41—21no—
263#41#21512yes—
264#41#21612yes—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8#21711yescharacteristic
268C2×D4—21no—
269C2×D4#21711yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—121no—
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277M16—121no—
278M16—121no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—62no—
286C2×C8—122no—
287C2×C8—124no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—62no—
298C4⋊C4—121no—
299C4⋊C4#21711yescharacteristic
300C4⋊C4#21711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—41no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—121no—
310C42—121no—
311C42D3×D411yescharacteristic
312C42#23711yescharacteristic
313C42C22×Dic311yescharacteristic
314C42C2×#6711yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3#38611yescharacteristic
321C22×C3#46511yescharacteristic
322C22×C3#46711yescharacteristic
323C3×C4—21no—
324C3×C4—22no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4C2×#10311yescharacteristic
332C3×C4#41611yescharacteristic
333C3×C4#44311yescharacteristic
334C3×C4#44411yescharacteristic
335C3×C4#46511yescharacteristic
336C3×C4#46711yescharacteristic
337Dic3—82no—
338C23—21no—
339C23#93911yescharacteristic
340Q8—21no—
341Q8—41no—
342Q8—41no—
343Q8—81no—
344Q8#83512yes—
345D4—22no—
346D4—22no—
347D4—21no—
348D4#83512yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—122no—
363C2×C4—241no—
364C2×C4—241no—
365C2×C4D4×Dic311yescharacteristic
366C2×C4D4×Dic311yescharacteristic
367C2×C4#93511yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#93911yescharacteristic
370C2×C4C2×#21511yescharacteristic
371C2×C4C2×#21611yescharacteristic
372C2×C4#95011yescharacteristic
373C2×C4#95011yescharacteristic
374C2×C4C2×#21711yescharacteristic
375C8—41no—
376C8—41no—
377C8—62no—
378C8—121no—
379C8—122no—
380C8—241no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3#187611yescharacteristic
385C2×C3#189211yescharacteristic
386C2×C3#190611yescharacteristic
387C22—21no—
388C22—21no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22#504811yescharacteristic
393C22#505111yescharacteristic
394C22#522011yescharacteristic
395C4—22no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—242no—
404C4#504811yescharacteristic
405C4#505111yescharacteristic
406C4#519611yescharacteristic
407C4#519711yescharacteristic
408C4C2×#83511yescharacteristic
409C4#523311yescharacteristic
410C3#920711yescharacteristic, complemented, Sylow
411C2—21no—
412C2—21no—
413C2—42no—
414C2#7169611yescharacteristic
415C2#7208711yescharacteristic
416C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3076C2
socleC22×C3#386
prime radicalC2×C3×#41D4
cosocleC2×C3×C42C23
hypercenter#3076D3
hypocenterC3#9207
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
3. #71835C2semidirect
4. #71826C2semidirect
6. #67707C2semidirect
7. #67704C2semidirect
13. C3×#477C4semidirect
14. C3×#476C4semidirect
15. C3×#475C4semidirect
16. C3×#474C4semidirect
94. #477Dic3semidirect
95. #476Dic3semidirect
96. #475Dic3semidirect
97. #474Dic3semidirect
410. C3#9207semidirect

Maximal subgroups

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

subgroupindex
C3×#30762
#41×C3⋊C82
#718352
#718262
#716462
#677072
#677042
#92073