Finite Group Databasebeta

← #10691070

Subgroups — #10691070

Order 768.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83209C211yesmaximal, characteristic
2#79040C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic, complemented
4C2×#4547C211yesmaximal, characteristic, complemented
5#70755C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9172—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#942—21no—
11C4×#235C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5225—21no—
15#5129—21no—
16#5122—21no—
17#5090—21no—
18#5090C412yes—
19#5083—21no—
20#5083C412yes—
21#4924C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4840—22no—
25#4835—21no—
26#4833—21no—
27C3×#332—22no—
28#4547—21no—
29#4547—21no—
30#4459—21no—
31#4456—21no—
32#3039—31no—
33#2925—31no—
34#1546D311yescharacteristic, 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
49C2×#216—22no—
50#942—42no—
51#942—42no—
52C2×#215—22no—
53C4×#67—21no—
54C4×#67—21no—
55#926D411yescharacteristic
56C2×#208—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C4×D12C2×C411yescharacteristic
60C4×D12D411yescharacteristic
61#872—21no—
62C42×D3—21no—
63C4×Dic6C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70#835—22no—
71#835—22no—
72(C2×D3)⋊C8—22no—
73(C2×D3)⋊C8—22no—
74#815—22no—
75C8×Dic3—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yes—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83C2×#137—61no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C4×#41—31no—
86#462—32no—
87#462—61no—
88#461—32no—
89#461—61no—
90#433—62no—
91#432—61no—
92#431—61no—
93#431C2×D311yescharacteristic
94#423—31no—
95#423C2×D311yescharacteristic
96C2×C4⋊C8—31no—
97C2×C4⋊C8—31no—
98#419—61no—
99#408—61no—
100#405—61no—
101#332—22no—
102#332—61no—
103#332—61no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C2×#67—21no—
117#235—21no—
118#235—21no—
119#235—21no—
120#235—21no—
121C2×D12C22⋊C411yescharacteristic
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×Dic6C22⋊C411yescharacteristic
126C3×M16—41no—
127C3×M16—41no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138#217—21no—
139#216—44no—
140#215—44no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3C22⋊C411yescharacteristic
144C4⋊Dic3C22⋊C411yescharacteristic
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149C4×Dic3—21no—
150#208—41no—
151#208—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—81no—
159C2×#41—31no—
160C22×Q8—21no—
161#137—122no—
162#137—122no—
163C2×Q16—62no—
164C2×SD16—62no—
165C2×M16—21no—
166C2×M16—61no—
167C22×C8—61no—
168C22×C8—61no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C4×D312yes—
173C4×Q8—31no—
174C4×Q8—31no—
175C4×D4—31no—
176C4×D4—31no—
177C4×D4—61no—
178C4×D4—61no—
179#117—31no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—31no—
189C4⋊C8—31no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8#6712yes—
195#103—62no—
196#103—62no—
197C22⋊C8—21no—
198C22⋊C8—62no—
199C22⋊C8—62no—
200#97—62no—
201C4×C8—21no—
202C4×C8—62no—
203C23×C3#12311yescharacteristic
204C22×D3—21no—
205C3×Q8—42no—
206C3×Q8—42no—
207C3×Q8—81no—
208C3×Q8—81no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225#67—41no—
226#67—41no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230D12—21no—
231D12—21no—
232C4×D3—41no—
233C4×D3—41no—
234C4×D3—41no—
235C4×D3—41no—
236C4×D3—81no—
237Dic6—21no—
238Dic6—21no—
239C3×C8—42no—
240C3×C8—81no—
241C3⋊C8—42no—
242C3⋊C8—41no—
243C3⋊C8—41no—
244C3⋊C8—81no—
245#41—61no—
246#41—61no—
247#41—61no—
248#41—61no—
249C2×Q8—22no—
250C2×Q8—31no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—31no—
257C2×D4—61no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4#23511yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4#23711yescharacteristic
265Q16—124no—
266SD16—124no—
267M16—41no—
268M16—41no—
269M16—121no—
270M16—121no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—62no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—121no—
277C2×C8—121no—
278C2×C8—121no—
279C2×C8—241no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C42—61no—
292C42—61no—
293C42—61no—
294C42C2×C4×D311yescharacteristic
295C42D3×D411yescharacteristic
296C42D3×D411yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#38611yescharacteristic
301C22×C3#48211yescharacteristic
302C22×C3#48211yescharacteristic
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—82no—
310C3×C4#41611yescharacteristic
311C3×C4#41811yescharacteristic
312C3×C4#44911yescharacteristic
313C3×C4#44911yescharacteristic
314C3×C4#47111yescharacteristic
315C3×C4#47311yescharacteristic
316Dic3—41no—
317Dic3—41no—
318Dic3—41no—
319C23—61no—
320C23#88311yescharacteristic
321Q8—42no—
322Q8—42no—
323Q8—61no—
324Q8—61no—
325Q8—81no—
326Q8—81no—
327D4—61no—
328D4—61no—
329D4—121no—
330D4—121no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—241no—
345C2×C4D3×C22⋊C411yescharacteristic
346C2×C4#67⋊C411yescharacteristic
347C2×C4#88411yescharacteristic
348C2×C4#91111yescharacteristic
349C2×C4#91211yescharacteristic
350C2×C4#91411yescharacteristic
351C2×C4#91611yescharacteristic
352C2×C4C4×#6711yescharacteristic
353C2×C4#93711yescharacteristic
354C2×C4#93811yescharacteristic
355C8—42no—
356C8—81no—
357C8—121no—
358C8—121no—
359C8—122no—
360C8—241no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3#187511yescharacteristic
364C2×C3#190211yescharacteristic
365C2×C3#190811yescharacteristic
366D3—81no—
367C22—21no—
368C22—21no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22#477911yescharacteristic
373C22#478411yescharacteristic
374C22#495611yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4#475411yescharacteristic
383C4#475511yescharacteristic
384C4#477011yescharacteristic
385C4#477211yescharacteristic
386C4#515811yescharacteristic
387C4#517211yescharacteristic
388C3#917211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—241no—
392C2#7012311yescharacteristic
393C2#7020111yescharacteristic
394C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9172
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
3. C3×#1546C2semidirect
4. C2×#4547C2semidirect
6. #67745C2semidirect
7. #67711C2semidirect
34. #1546D3semidirect
388. C3#9172semidirect

Maximal subgroups

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

subgroupindex
#832092
#790402
C3×#15462
C2×#45472
#707552
#677452
#677112
#91723