Finite Group Databasebeta

← #10691042

Subgroups — #10691042

Order 768.

432 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83212C211yesmaximal, characteristic
2#79040C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4C2×#4547C211yesmaximal, characteristic, complemented
5#70755C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9168—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5129—21no—
14#5122—21no—
15#5094—21no—
16#5091C412yes—
17#5087—21no—
18#5084C412yes—
19#4966—21no—
20#4964—21no—
21#4924C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4840—22no—
25#4835—21no—
26#4833—21no—
27C3×#333—22no—
28#4547—21no—
29#4547—21no—
30#4459—21no—
31#4456—21no—
32#3079—31no—
33#2925—31no—
34#1548D311yescharacteristic, complemented
35C2×#332—31no—
36#1477—31no—
37#1301—31no—
38#1292—31no—
39C22×C3×Q8—21no—
40C2×#235D411yescharacteristic
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8C2×C412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C4×#67—21no—
50C4×#67—21no—
51#926D411yescharacteristic
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8—21no—
55C4×D12C2×C411yescharacteristic
56C4×D12D411yescharacteristic
57#872—21no—
58C42×D3—21no—
59C4×Dic6C2×C411yescharacteristic
60C4×Dic6D411yescharacteristic
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8D412yes—
64C3×C22⋊C8—21no—
65C3×C4×C8—21no—
66(C2×D3)⋊C8—22no—
67(C2×D3)⋊C8—22no—
68#815—22no—
69C8×Dic3—22no—
70#810—22no—
71#809—22no—
72#808—22no—
73#807—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)C2×C412yes—
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D411yescharacteristic
81C2×C4⋊Q8C2×D311yescharacteristic
82C4×#41—31no—
83#477—32no—
84#476—32no—
85#475—61no—
86#474—61no—
87#433—62no—
88#432—61no—
89#431—61no—
90#431C2×D311yescharacteristic
91#428—61no—
92#423—31no—
93#423C2×D311yescharacteristic
94C2×C4⋊C8—31no—
95C2×C4⋊C8—31no—
96#417—61no—
97#408—61no—
98#405—61no—
99#333—22no—
100#332—61no—
101#332—61no—
102Q8⋊C8—61no—
103D4⋊C8—61no—
104C2×C3×Q8—22no—
105C2×C3×Q8—41no—
106C2×C3×Q8—81no—
107C2×C3×Q8—81no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—21no—
113#235—21no—
114#235—21no—
115#235—21no—
116#235—21no—
117C2×D12C22⋊C411yescharacteristic
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×Dic6C22⋊C411yescharacteristic
122C3×M16—41no—
123C3×M16—41no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134#217—21no—
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137C4⋊Dic3C22⋊C411yescharacteristic
138C4⋊Dic3C22⋊C411yescharacteristic
139Dic3⋊C4—21no—
140Dic3⋊C4—21no—
141C4×Dic3—21no—
142C4×Dic3—21no—
143C4×Dic3—21no—
144#208—41no—
145#208—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—81no—
153C2×#41—31no—
154C22×Q8—21no—
155C2×M16—21no—
156C2×M16—61no—
157C22×C8—61no—
158C22×C8—61no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8C4×D312yes—
163C4×Q8—31no—
164C4×Q8—31no—
165C4×D4—31no—
166C4×D4—31no—
167C4×D4—61no—
168C4×D4—61no—
169#117—31no—
170#117—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C42—61no—
174C2×C42C22×D311yescharacteristic
175#107—62no—
176#106—62no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—31no—
181C4⋊C8—31no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186C4⋊C8#6712yes—
187#103—62no—
188#102—62no—
189C22⋊C8—21no—
190C22⋊C8—62no—
191C22⋊C8—62no—
192#97—62no—
193C4×C8—21no—
194C4×C8—62no—
195C23×C3#12311yescharacteristic
196C22×D3—21no—
197C3×Q8—82no—
198C3×Q8—81no—
199C3×Q8—81no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—82no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4#13711yescharacteristic
214C2×C3×C4#13711yescharacteristic
215#67—41no—
216#67—41no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220D12—21no—
221D12—21no—
222C4×D3—41no—
223C4×D3—41no—
224C4×D3—41no—
225C4×D3—41no—
226C4×D3—81no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—42no—
232C3⋊C8—41no—
233C3⋊C8—41no—
234C3⋊C8—81no—
235#41—61no—
236#41—61no—
237#41—61no—
238#41—61no—
239C2×Q8—22no—
240C2×Q8—31no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×D4—31no—
245C2×D4—61no—
246C22×C4—41no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4#23511yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4#23711yescharacteristic
253M16—41no—
254M16—41no—
255M16—121no—
256M16—121no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—121no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—241no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C42—61no—
278C42—61no—
279C42—61no—
280C42C2×C4×D311yescharacteristic
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#38611yescharacteristic
287C22×C3#48511yescharacteristic
288C22×C3#48511yescharacteristic
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—82no—
296C3×C4#41611yescharacteristic
297C3×C4#41911yescharacteristic
298C3×C4#45211yescharacteristic
299C3×C4#45211yescharacteristic
300C3×C4#47111yescharacteristic
301C3×C4#47211yescharacteristic
302Dic3—41no—
303Dic3—41no—
304Dic3—41no—
305C23—61no—
306C23#88311yescharacteristic
307Q8—61no—
308Q8—61no—
309Q8—82no—
310Q8—81no—
311Q8—81no—
312D4—61no—
313D4—61no—
314D4—121no—
315D4—121no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—82no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4D3×C22⋊C411yescharacteristic
330C2×C4#67⋊C411yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#91511yescharacteristic
333C2×C4#91611yescharacteristic
334C2×C4#91811yescharacteristic
335C2×C4#91911yescharacteristic
336C2×C4C4×#6711yescharacteristic
337C2×C4#93711yescharacteristic
338C2×C4#93811yescharacteristic
339C8—42no—
340C8—81no—
341C8—121no—
342C8—121no—
343C8—122no—
344C8—241no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#187311yescharacteristic
348C2×C3#190211yescharacteristic
349C2×C3#190711yescharacteristic
350D3—81no—
351C22—21no—
352C22—21no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22#481411yescharacteristic
357C22#481511yescharacteristic
358C22#495611yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—121no—
364C4—121no—
365C4—121no—
366C4#474811yescharacteristic
367C4#474911yescharacteristic
368C4#480211yescharacteristic
369C4#480411yescharacteristic
370C4#517311yescharacteristic
371C4#518711yescharacteristic
372C3#916811yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—241no—
376C2#7008811yescharacteristic
377C2#7026911yescharacteristic
378C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9168
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. C2×#4547C2semidirect
6. #67745C2semidirect
7. #67711C2semidirect
34. #1548D3semidirect
372. C3#9168semidirect

Maximal subgroups

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

subgroupindex
#832122
#790402
C3×#15482
C2×#45472
#707552
#677452
#677112
#91683