Finite Group Databasebeta

← #10691961

Subgroups — #10691961

Order 768.

490 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82441C211yesmaximal, characteristic
2#82435C211yesmaximal, characteristic, complemented
3C3×#1805C211yesmaximal, characteristic, complemented
4#70426C211yesmaximal, characteristic
5#69545C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9306—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—21no—
14C3×#382C2211yescharacteristic
15#5031C412yescomplemented
16#5028—22no—
17#5027C412yescomplemented
18Q8×C3⋊C8—22no—
19Q8×C3⋊C8—22no—
20#4966—21no—
21#4925—22no—
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C3×#340—21no—
26#4483—22no—
27#4451—22no—
28#3037—31no—
29#2982—31no—
30#1805D311yescharacteristic, complemented
31#1517—31no—
32#1480—31no—
33#1301—31no—
34#1286—31no—
35C22×C3×Q8D411yescharacteristic
36C2×#235—21no—
37C22×C3×C8—21no—
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8C2×C412yes—
41C3×C4×Q8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—22no—
48#942—42no—
49#942—42no—
50C2×#215—22no—
51#930—21no—
52#930—21no—
53C2×#208—21no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869C2×C411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#103—42no—
62C3×#95—21no—
63C3×#95—21no—
64#830—21no—
65#830D412yes—
66#810—22no—
67#809—22no—
68C4⋊(C3⋊C8)—24no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)C2×C412yes—
73C4×C3⋊C8—24no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C2×#137—61no—
77#532—31no—
78C2×C4×Q8C2×D311yescharacteristic
79#464—62no—
80#462—32no—
81#460—32no—
82C8×Q8—62no—
83C8×Q8—62no—
84#431—31no—
85#431—31no—
86#431—62no—
87#423—31no—
88C2×C4⋊C8C2×D311yescharacteristic
89#417—61no—
90C2×#103—21no—
91#382C2×D311yescharacteristic
92#340—21no—
93#327—62no—
94D4⋊C8—62no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8C22⋊C412yes—
99C2×C3×Q8D812yes—
100C2×C3×Q8SD1612yes—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—41no—
108#235—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×C4×D3—41no—
116C2×Dic6—21no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4C22⋊C412yes—
126C3×C42—22no—
127C3×C42—22no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#216—44no—
133#215—44no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138#208—41no—
139#208—41no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×#41—61no—
144C22×Q8#6711yescharacteristic
145#137—122no—
146#137—122no—
147C2×Q16—62no—
148C2×SD16—62no—
149C2×M16—61no—
150C22×C8—21no—
151C4⋊Q8—31no—
152#127—31no—
153#124—61no—
154#121—61no—
155#121—61no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×Q8C4×D312yes—
159C4×Q8D1212yes—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4#6711yescharacteristic
165C2×C42—21no—
166C2×C42C22×D311yescharacteristic
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—32no—
170C4⋊C8—64no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174#103—42no—
175#103—62no—
176#102—62no—
177C22⋊C8—32no—
178C22⋊C8—61no—
179C4×C8—32no—
180C4×C8—64no—
181C4×C8—61no—
182#95—21no—
183#95—21no—
184C23×C3#12111yescharacteristic
185C22×D3—41no—
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—22no—
189C3×Q8#10214yes—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4C2×D811yescharacteristic
207C2×C3×C4C2×SD1611yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×C3×C4#13511yescharacteristic
210#67—81no—
211#67—81no—
212C2×Dic3—41no—
213D12—42no—
214D12—41no—
215D12—41no—
216D12—81no—
217C4×D3—81no—
218C4×D3—81no—
219Dic6—41no—
220Dic6—41no—
221C3×C8—81no—
222C3⋊C8—42no—
223C3⋊C8—42no—
224C3⋊C8—42no—
225#41—121no—
226#41—121no—
227#41—121no—
228#41—121no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—61no—
233C2×Q8(C2×D3)⋊C412yes—
234C2×Q8#21312yes—
235C2×Q8#21512yes—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—121no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—121no—
244C22×C4D3×D411yescharacteristic
245C22×C4#23711yescharacteristic
246C22×C4C2×#6711yescharacteristic
247Q16—124no—
248SD16—124no—
249M16—121no—
250M16—121no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—62no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4(C2×D3)⋊C412yes—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C42—22no—
268C42—22no—
269C42C2×C4×D311yescharacteristic
270C42C2×D1211yescharacteristic
271C42D3×D411yescharacteristic
272C42#23711yescharacteristic
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#38611yescharacteristic
276C22×C3#46511yescharacteristic
277C22×C3#46611yescharacteristic
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—24no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—161no—
288C3×C4C2×#10211yescharacteristic
289C3×C4#41611yescharacteristic
290C3×C4#46511yescharacteristic
291C3×C4#46611yescharacteristic
292Dic3—81no—
293C23—121no—
294C23#93911yescharacteristic
295Q8—22no—
296Q8—22no—
297Q8—22no—
298Q8—121no—
299Q8—121no—
300Q8#80914yes—
301D4—122no—
302D4—121no—
303D4—121no—
304D4—241no—
305D4—241no—
306D4—241no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—121no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4#67⋊C411yescharacteristic
321C2×C4#67⋊C411yescharacteristic
322C2×C4#88211yescharacteristic
323C2×C4#88611yescharacteristic
324C2×C4C2×(C2×D3)⋊C411yescharacteristic
325C2×C4C2×#21311yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4C2×#21511yescharacteristic
328C2×C4#95011yescharacteristic
329C2×C4#95011yescharacteristic
330C8—81no—
331C8—122no—
332C8—122no—
333C8—122no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#188111yescharacteristic
337C2×C3#189311yescharacteristic
338C2×C3#190511yescharacteristic
339D3—161no—
340C22—21no—
341C22—21no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22#495811yescharacteristic
346C22#503811yescharacteristic
347C22#504811yescharacteristic
348C4—24no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—161no—
355C4—241no—
356C4C2×#80911yescharacteristic
357C4#500211yescharacteristic
358C4#503811yescharacteristic
359C4#504811yescharacteristic
360C3#930611yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—481no—
364C2#6951511yescharacteristic
365C2#7029311yescharacteristic
366C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1805C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1805D3
hypocenterC3#9306
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. #82435C2semidirect
3. C3×#1805C2semidirect
6. #67707C2semidirect
7. #67631C2semidirect
15. #5031C4semidirect
17. #5027C4semidirect
30. #1805D3semidirect
360. C3#9306semidirect

Maximal subgroups

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

subgroupindex
#824412
#824352
C3×#18052
#704262
#695452
#677072
#676312
#93063