Finite Group Databasebeta

← #10691072

Subgroups — #10691072

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#79057C211yesmaximal, characteristic
2#79000C211yesmaximal, characteristic
3#70778C211yesmaximal, characteristic, complemented
4#70755C211yesmaximal, characteristic
5C2×#4482C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9172—31nomaximal, Sylow
9C2×#909—21no—
10C4×#235C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5120—21no—
16#5110—21no—
17#5101—21no—
18#4924C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4849—21no—
21#4849C412yes—
22#4841—21no—
23#4841C412yes—
24#4840—22no—
25#4835—21no—
26#4833—21no—
27#4486—22no—
28#4482—21no—
29#4482—21no—
30#4459—21no—
31#4456—21no—
32#3039—31no—
33#2925—31no—
34#1546—31no—
35C2×#332—31no—
36#1477—31no—
37#1301D311yescharacteristic, complemented
38#1292—31no—
39C2×#235D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44C4×#67—21no—
45C4×#67—21no—
46#926D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51#909—42no—
52#909—42no—
53C2×Dic12—22no—
54C2×#204—22no—
55C4×D12C2×C411yescharacteristic
56C4×D12D411yescharacteristic
57#872—21no—
58C42×D3—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869C2×C412yes—
63C4×Dic6C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×C4⋊C8D411yescharacteristic
70C3×C4⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73(C2×D3)⋊C8—22no—
74(C2×D3)⋊C8—22no—
75#816—22no—
76#816—22no—
77#815—22no—
78C8×Dic3—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C2×#137—61no—
84C2×C4⋊Q8—31no—
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⋊C8C2×D311yescharacteristic
98#419—61no—
99#408—61no—
100#405—61no—
101#332—62no—
102#332—61no—
103#332—61no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—21no—
110C22×Dic3—41no—
111#235—21no—
112#235—21no—
113#235—21no—
114#235—21no—
115C2×D12C22⋊C411yescharacteristic
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×Dic6—22no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6C22⋊C411yescharacteristic
126C3×M16—41no—
127C3×M16—41no—
128C2×C3×C8—22no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138#217—21no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3C22⋊C411yescharacteristic
146C4⋊Dic3C22⋊C411yescharacteristic
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—81no—
157Dic12—44no—
158#204—44no—
159C2×#41—31no—
160C22×Q8—61no—
161#137—122no—
162#137—122no—
163C2×Q16—62no—
164C2×SD16—62no—
165C2×M16—21no—
166C2×M16—61no—
167C22×C8—21no—
168C22×C8—61no—
169C4⋊Q8—32no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172C4⋊Q8—61no—
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—61no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8C4×D312yes—
193C4⋊C8D1211yescharacteristic
194C4⋊C8D1211yescharacteristic
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—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218#67—41no—
219#67—41no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—42no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226D12—21no—
227D12—21no—
228C4×D3—41no—
229C4×D3—41no—
230C4×D3—41no—
231C4×D3—41no—
232C4×D3—81no—
233Dic6—21no—
234Dic6—21no—
235Dic6—42no—
236Dic6—42no—
237Dic6—81no—
238Dic6—81no—
239C3×C8—42no—
240C3×C8—41no—
241C3×C8—41no—
242C3×C8—81no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245#41—61no—
246#41—61no—
247#41—61no—
248#41—61no—
249C2×Q8—31no—
250C2×Q8—62no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×D4—31no—
257C2×D4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—121no—
262C22×C4#23511yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4#23911yescharacteristic
265Q16—124no—
266SD16—124no—
267M16—41no—
268M16—41no—
269M16—121no—
270M16—121no—
271C2×C8—22no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—62no—
278C2×C8—121no—
279C2×C8—241no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C42—61no—
292C42—61no—
293C42—61no—
294C42C2×C4×D311yescharacteristic
295C42C2×D1211yescharacteristic
296C42D3×D411yescharacteristic
297C42D3×D411yescharacteristic
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#41611yescharacteristic
310C3×C4#41811yescharacteristic
311C3×C4#44911yescharacteristic
312C3×C4#44911yescharacteristic
313C3×C4#47111yescharacteristic
314C3×C4#47311yescharacteristic
315Dic3—41no—
316Dic3—41no—
317Dic3—41no—
318Dic3—82no—
319C23—61no—
320C23#89411yescharacteristic
321Q8—61no—
322Q8—61no—
323Q8—122no—
324Q8—122no—
325Q8—241no—
326Q8—241no—
327D4—61no—
328D4—61no—
329D4—121no—
330D4—121no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4—241no—
345C2×C4C4×D1211yescharacteristic
346C2×C4D3×C22⋊C411yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#89311yescharacteristic
350C2×C4#89511yescharacteristic
351C2×C4#91111yescharacteristic
352C2×C4#91411yescharacteristic
353C2×C4#91611yescharacteristic
354C2×C4#91911yescharacteristic
355C8—42no—
356C8—41no—
357C8—41no—
358C8—81no—
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#466511yescharacteristic
373C22#486611yescharacteristic
374C22#488511yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—121no—
379C4—121no—
380C4—121no—
381C4—242no—
382C4#477011yescharacteristic
383C4#478111yescharacteristic
384C4#480411yescharacteristic
385C4#481011yescharacteristic
386C4#489611yescharacteristic
387C4#489811yescharacteristic
388C3#917211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—241no—
392C2#6888111yescharacteristic
393C2#6894911yescharacteristic
394C2#6903711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9172
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70778C2semidirect
5. C2×#4482C2semidirect
6. C3×#1301C2semidirect
7. #67711C2semidirect
37. #1301D3semidirect
388. C3#9172semidirect

Maximal subgroups

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

subgroupindex
#790572
#790002
#707782
#707552
C2×#44822
C3×#13012
#677112
#91723