Finite Group Databasebeta

← #10691581

Subgroups — #10691581

Order 768.

528 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
1C2×#5087C211yesmaximal, characteristic
2M16×D12C211yesmaximal, characteristic
3C3×#1966C211yesmaximal, characteristic, complemented
4#71773C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic
6#67711C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9251—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×D3×M16—21no—
11C2×C4×D12C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#101—21no—
14C2×C3×#101—21no—
15C3×C4×M16C2211yescharacteristic
16#5126—21no—
17#5087—21no—
18#5087—21no—
19#5087—21no—
20#5087—21no—
21C2×#809C412yes—
22C2×#807C412yes—
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4706—22no—
27#4688—21no—
28C8×D12—21no—
29#4547—22no—
30#4483—21no—
31#4483—21no—
32C2×#474—31no—
33D4×M16—31no—
34#1966D311yescharacteristic, complemented
35#1546—31no—
36#1486—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3—21no—
42C2×C3×M16—21no—
43C2×C3×M16D411yescharacteristic
44C2×C3×M16D411yescharacteristic
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4C2×C412yes—
51C2×C3×C42C2311yescharacteristic
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×#208—21no—
55C22×C3⋊C8C2×C412yes—
56D3×M16—42no—
57D3×M16—42no—
58C2×ΓL(1,𝔽25)—22no—
59C2×D3×C8—22no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12D411yescharacteristic
63C4×D12D411yescharacteristic
64C4×D12D411yescharacteristic
65C4×D12D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#101—42no—
69C3×#101—42no—
70C3×#97—21no—
71C3×C4×C8—21no—
72#830—21no—
73(C2×D3)⋊C8—22no—
74(C2×D3)⋊C8—22no—
75#809—22no—
76#809—22no—
77#807—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C22×M16—61no—
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×C4×D4—31no—
88#474—61no—
89#474—61no—
90#474—61no—
91#474—61no—
92#436—61no—
93#435—62no—
94C8×D4—61no—
95#431—31no—
96C2×#107—32no—
97#423—31no—
98#423C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100C2×#102—32no—
101C2×#101—21no—
102C2×#101—21no—
103#407—61no—
104C4×M16C2×D311yescharacteristic
105#332—62no—
106D4⋊C8—61no—
107D4⋊C8—61no—
108C23×D3—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C22×C3×C4—22no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C2×D12—22no—
121C2×D12—21no—
122C2×D12—41no—
123C2×D12C22⋊C411yescharacteristic
124C2×D12C22⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C3×M16—21no—
134C3×M16—21no—
135C3×M16—21no—
136C3×M16—41no—
137C3×M16—41no—
138C3×M16D812yes—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C2×C3×C8—41no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150(C2×D3)⋊C4—21no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3C22⋊C411yescharacteristic
156C4⋊Dic3C22⋊C411yescharacteristic
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C2×C3⋊C8—41no—
163ΓL(1,𝔽25)—44no—
164D3×C8—44no—
165C22×Q8—21no—
166C22×D4—31no—
167C23×C4—61no—
168C2×M16—21no—
169C2×M16—62no—
170C2×M16—61no—
171C2×M16—122no—
172C2×M16—122no—
173C2×M16#6711yescharacteristic
174C2×M16#6711yescharacteristic
175C22×C8—32no—
176C22×C8—62no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4×D4—31no—
182C4×D4—31no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—61no—
186C4×D4—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4C4×D312yes—
190C2×C22⋊C4—61no—
191C2×C42C22×D311yescharacteristic
192#107—62no—
193#107—62no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#102—62no—
202#102—62no—
203#101—42no—
204#101—42no—
205C22⋊C8—62no—
206C22⋊C8—62no—
207C22⋊C8—61no—
208#97—21no—
209C4×C8—21no—
210C4×C8—61no—
211C23×C3C4×D411yescharacteristic
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—42no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—81no—
218C3×Q8—81no—
219C3×Q8—81no—
220C3×Q8—81no—
221C3×Q8—81no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242D12—21no—
243D12—21no—
244D12—21no—
245D12—42no—
246D12#10112yescomplemented
247C4×D3—41no—
248C4×D3—41no—
249C4×D3—41no—
250C4×D3—41no—
251C4×D3—41no—
252C4×D3—41no—
253C4×D3—41no—
254C4×D3—41no—
255C3×C8—22no—
256C3×C8—41no—
257C3×C8—81no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C24—61no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—31no—
269C2×D4—31no—
270C2×D4—62no—
271C2×D4—61no—
272C2×D4—121no—
273C22×C4—22no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4C2×#6711yescharacteristic
286M16—21no—
287M16—21no—
288M16—21no—
289M16—41no—
290M16—41no—
291M16—124no—
292M16—121no—
293M16—121no—
294M16#21312yes—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—124no—
301C2×C8—122no—
302C2×C8—121no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C42D3×D411yescharacteristic
315C42D3×D411yescharacteristic
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3#38611yescharacteristic
321C22×C3C4×D811yescharacteristic
322C22×C3#44211yescharacteristic
323C2×D3—22no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—82no—
330C2×D3—82no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—82no—
335C3×C4C2×#10111yescharacteristic
336C3×C4#41311yescharacteristic
337C3×C4#44711yescharacteristic
338C3×C4#44911yescharacteristic
339C3×C4#45911yescharacteristic
340C3×C4#46311yescharacteristic
341Dic3—41no—
342Dic3—41no—
343C23—61no—
344C23—61no—
345C23—122no—
346C23—121no—
347C23—121no—
348C23—241no—
349C23C4×#6711yescharacteristic
350Q8—81no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354D4—32no—
355D4—61no—
356D4—61no—
357D4—61no—
358D4—122no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4D3×C22⋊C411yescharacteristic
378C2×C4#67⋊C411yescharacteristic
379C2×C4#88311yescharacteristic
380C2×C4#88411yescharacteristic
381C2×C4C2×#21311yescharacteristic
382C2×C4#93211yescharacteristic
383C2×C4#93711yescharacteristic
384C2×C4#93811yescharacteristic
385C2×C4#94911yescharacteristic
386C2×C4#95011yescharacteristic
387C8—22no—
388C8—41no—
389C8—81no—
390C8—122no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3#187911yescharacteristic
396C2×C3#190111yescharacteristic
397C2×C3#190211yescharacteristic
398D3—42no—
399D3—81no—
400C22—21no—
401C22—21no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—242no—
409C22—242no—
410C22#495611yescharacteristic
411C22C4×#21311yescharacteristic
412C22#501411yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—82no—
417C4—121no—
418C4—121no—
419C4D3×#10111yescharacteristic
420C4#475111yescharacteristic
421C4#503611yescharacteristic
422C4#503711yescharacteristic
423C4#505711yescharacteristic
424C4#507211yescharacteristic
425C3#925111yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—122no—
429C2—241no—
430C2#6965711yescharacteristic
431C2#7011511yescharacteristic
432C2#7012211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1966C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1966D3
hypocenterC3#9251
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
3. C3×#1966C2semidirect
4. #71773C2semidirect
6. #67711C2semidirect
7. #67668C2semidirect
34. #1966D3semidirect
246. D12#101semidirect
425. C3#9251semidirect

Maximal subgroups

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

subgroupindex
C2×#50872
M16×D122
C3×#19662
#717732
#704962
#677112
#676682
#92513