Finite Group Databasebeta

← #10691078

Subgroups — #10691078

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82526C211yesmaximal, characteristic
2#82512C211yesmaximal, characteristic, complemented
3C3×#1477C211yesmaximal, characteristic, complemented
4#70774C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9173—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#949—21no—
11C2×#948—21no—
12C2×#874C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5027—21no—
16#5027C412yes—
17#5015—21no—
18#5015C412yes—
19#5001—21no—
20#4999—21no—
21#4999—22no—
22#4924—21no—
23#4924C2211yescharacteristic
24#4924C2211yescharacteristic
25C2×C4⋊(C3⋊C8)—21no—
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×Q8⋊C8—21no—
28C3×D4⋊C8—21no—
29#4485—21no—
30#4485—21no—
31#4485—22no—
32#3038—31no—
33#2906—31no—
34#1545—31no—
35C2×#331—31no—
36#1477D311yescharacteristic, complemented
37#1301—31no—
38#1292—31no—
39C2×C3×#41D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×C3×C8—21no—
44C3×C4×Q8C2×C411yescharacteristic
45C3×C4×Q8D411yescharacteristic
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×#117—21no—
51C3×#117D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54#949—42no—
55#949—42no—
56#948—42no—
57#948—42no—
58C2×#215—22no—
59C2×#213—22no—
60C2×#208—22no—
61C2×#208—21no—
62C2×#208—21no—
63C22×C3⋊C8—22no—
64C22×C3⋊C8—21no—
65#874—21no—
66#874—21no—
67#874—21no—
68#874C2×C412yes—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#809—22no—
75#809—22no—
76C4⋊(C3⋊C8)—22no—
77C4⋊(C3⋊C8)—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)C2×C412yes—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84C4⋊(C3⋊C8)D412yes—
85C2×AGL(1,ℤ/8)—61no—
86C2×#131—61no—
87C2×#127—31no—
88C4×#41C2×D311yescharacteristic
89#460—32no—
90#460—61no—
91#459—32no—
92#459—61no—
93#431C2×D311yescharacteristic
94#424—62no—
95#424—61no—
96#423—31no—
97#423—31no—
98#423—61no—
99C2×C4⋊C8—31no—
100C2×C4⋊C8—61no—
101C2×C4⋊C8C2×D311yescharacteristic
102#418—61no—
103#331—61no—
104#331—61no—
105#331—62no—
106Q8⋊C8—21no—
107D4⋊C8—21no—
108C23×D3—41no—
109C3×#41—21no—
110C3×#41—21no—
111C3×#41D812yes—
112C3×#41SD1612yes—
113C2×C3×Q8C22⋊C411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4C22⋊C411yescharacteristic
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×D12—22no—
123C2×D12—22no—
124C2×D12—22no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—82no—
133C2×D12—81no—
134C2×D12—81no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—81no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C4⋊C4C22⋊C411yescharacteristic
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C42—21no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#215—44no—
153#213—44no—
154#208—44no—
155#208—41no—
156#208—41no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—44no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41D1211yescharacteristic
167C22×D4—61no—
168C22×D4—61no—
169C22×D4—61no—
170AGL(1,ℤ/8)—122no—
171AGL(1,ℤ/8)—122no—
172C2×SD16—62no—
173C2×D8—62no—
174#131—122no—
175#131—122no—
176C2×M16—62no—
177C2×M16—61no—
178C2×M16—61no—
179C22×C8—21no—
180C22×C8—62no—
181C22×C8—61no—
182#127—32no—
183#127—61no—
184#127—61no—
185#127—61no—
186C4×Q8C4×D311yescharacteristic
187C4×Q8#6711yescharacteristic
188C4×D4—21no—
189C4×D4—21no—
190C4×D4C4×D311yescharacteristic
191C4×D4#6711yescharacteristic
192#117—21no—
193#117D1211yescharacteristic
194C2×C42—21no—
195C2×C42C22×D311yescharacteristic
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—32no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202C4⋊C8—32no—
203C4⋊C8—62no—
204C4⋊C8—62no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C4×C8—21no—
212C23×C3—21no—
213C23×C3#12311yescharacteristic
214C22×D3—42no—
215C22×D3—82no—
216C22×D3—82no—
217C22×D3—81no—
218C22×D3—81no—
219C3×Q8—21no—
220C3×Q8#10212yes—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—21no—
224C3×D4#10212yes—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—81no—
236C2×C3×C4C2×C22⋊C411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4C2×D811yescharacteristic
243C2×C3×C4C2×SD1611yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×C3×C4#13711yescharacteristic
246D12—42no—
247D12—42no—
248D12—82no—
249D12—82no—
250D12—82no—
251D12—81no—
252D12—81no—
253D12—81no—
254D12—81no—
255C3×C8—81no—
256C3×C8—81no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—42no—
260C3⋊C8—41no—
261C24—121no—
262#41—21no—
263#41—21no—
264#41#20412yes—
265#41D2412yes—
266C2×Q8(C2×D3)⋊C411yescharacteristic
267C2×D4—21no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—241no—
279C2×D4—241no—
280C2×D4—242no—
281C2×D4(C2×D3)⋊C411yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4C2×D1211yescharacteristic
286C22×C4#23711yescharacteristic
287C22×C4#23711yescharacteristic
288SD16—124no—
289D8—124no—
290M16—124no—
291M16—121no—
292M16—121no—
293M16—121no—
294M16—121no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8—81no—
302C2×C8—124no—
303C2×C8—121no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4(C2×D3)⋊C411yescharacteristic
308C4⋊C4(C2×D3)⋊C411yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42C2×C4×D311yescharacteristic
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3#38611yescharacteristic
325C22×C3#48011yescharacteristic
326C22×C3#48111yescharacteristic
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—162no—
331C2×D3—162no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—41no—
337C3×C4—41no—
338C3×C4C2×#10211yescharacteristic
339C3×C4#41711yescharacteristic
340C3×C4#44711yescharacteristic
341C3×C4#44811yescharacteristic
342C3×C4#47111yescharacteristic
343C3×C4#47211yescharacteristic
344C23—21no—
345C23—122no—
346C23—242no—
347C23—242no—
348C23—241no—
349C23—241no—
350C23#88611yescharacteristic
351Q8—21no—
352Q8#82112yes—
353D4—22no—
354D4—22no—
355D4—21no—
356D4—122no—
357D4—122no—
358D4—242no—
359D4—242no—
360D4—242no—
361D4—241no—
362D4—241no—
363D4—241no—
364D4—241no—
365D4#82112yes—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4#67⋊C411yescharacteristic
378C2×C4#67⋊C411yescharacteristic
379C2×C4#88211yescharacteristic
380C2×C4C2×#20411yescharacteristic
381C2×C4C2×D2411yescharacteristic
382C2×C4#90811yescharacteristic
383C2×C4#90911yescharacteristic
384C2×C4C2×(C2×D3)⋊C411yescharacteristic
385C2×C4#93911yescharacteristic
386C2×C4#93911yescharacteristic
387C8—81no—
388C8—81no—
389C8—121no—
390C8—122no—
391C8—121no—
392C8—122no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—42no—
396C2×C3#187411yescharacteristic
397C2×C3#190311yescharacteristic
398C2×C3#190711yescharacteristic
399D3—162no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—242no—
406C22—242no—
407C22—242no—
408C22—482no—
409C22—482no—
410C22#473511yescharacteristic
411C22#473611yescharacteristic
412C22#495811yescharacteristic
413C4—21no—
414C4—21no—
415C4—22no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4#473211yescharacteristic
420C4#473811yescharacteristic
421C4C2×#82111yescharacteristic
422C4#513311yescharacteristic
423C4#519911yescharacteristic
424C4#520311yescharacteristic
425C3#917311yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—42no—
429C2—482no—
430C2#6993211yescharacteristic
431C2#7030311yescharacteristic
432C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1477C2
socleC22×C3#386
prime radical#82526C2
cosocleC2×C3×C42C23
hypercenter#1477D3
hypocenterC3#9173
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #82512C2semidirect
3. C3×#1477C2semidirect
4. #70774C2semidirect
5. C2×#4485C2semidirect
6. #67745C2semidirect
7. #67705C2semidirect
36. #1477D3semidirect
425. C3#9173semidirect

Maximal subgroups

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

subgroupindex
#825262
#825122
C3×#14772
#707742
C2×#44852
#677452
#677052
#91733