Finite Group Databasebeta

← #10692075

Subgroups — #10692075

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2989C211yesmaximal, characteristic, complemented
2#82526C211yesmaximal, characteristic
3#71646C211yesmaximal, characteristic, complemented
4#70774C211yesmaximal, characteristic, complemented
5#70457C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9322—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#949—21no—
11C2×#874C2211yescharacteristic
12C3×#446—21no—
13C3×#436—21no—
14C3×#435—22no—
15C3×#432—21no—
16C3×#431C2211yescharacteristic
17C3×#423—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×#837—21no—
20C2×#821—22no—
21#5027—21no—
22#5027C412yescomplemented
23#5015—21no—
24#5015C412yescomplemented
25#5001—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4924C2211yescharacteristic
28#4924C2211yescharacteristic
29#4532—21no—
30#4485—21no—
31#4485—21no—
32#3038—31no—
33#2989D311yescharacteristic, complemented
34#1982—31no—
35#1806—31no—
36#1545—31no—
37#1303—31no—
38#1292—31no—
39C2×C3×#41D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C2×C3×M16—22no—
44C22×C3×C8D412yes—
45C3×C4×Q8C2×C411yescharacteristic
46C3×C4×Q8D411yescharacteristic
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4C2×C411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C3×#117—21no—
52C3×#117D411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55#949—42no—
56#949—42no—
57C2×#215—22no—
58C2×#213—22no—
59C2×C4⋊Dic3—21no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62C2×#208—21no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874C2×C412yes—
67C3×C4⋊C8—22no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D411yescharacteristic
71C3×C4⋊C8D411yescharacteristic
72C3×C22⋊C8—22no—
73C3×C22⋊C8—21no—
74C3×#97—22no—
75C3×C4×C8—21no—
76#837—42no—
77C3⋊#95—21no—
78#821—44no—
79#809—22no—
80#809—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)C2×C412yes—
85C2×AGL(1,ℤ/8)—61no—
86C2×#127—31no—
87C4×#41C2×D311yescharacteristic
88#460—32no—
89#460—61no—
90#459—32no—
91#459—61no—
92#446—21no—
93#436—21no—
94#435—22no—
95#432—21no—
96#431C2×D311yescharacteristic
97#423—21no—
98#423—31no—
99#423—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#104—61no—
102#418—61no—
103C2×#102—62no—
104C4×C4⋊C4—31no—
105#339—61no—
106#331—61no—
107#331—61no—
108C23×D3—41no—
109C3×#41—21no—
110C3×#41—21no—
111C3×#41—21no—
112C3×#41—21no—
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×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—41no—
123C2×D12—22no—
124C2×D12—22no—
125C2×D12—22no—
126C2×D12—42no—
127C2×D12—42no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—82no—
136C3×M16—82no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—22no—
140C2×C3×C8—42no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42C22×C411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#215—44no—
156#213—44no—
157C4⋊Dic3—22no—
158C4⋊Dic3—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161#208—41no—
162#208—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×#41D1211yescharacteristic
168C22×D4—61no—
169C22×D4—61no—
170C22×D4—61no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×D8—62no—
175C2×M16—22no—
176C2×M16—61no—
177C2×M16—61no—
178C22×C8#6712yes—
179#127—32no—
180#127—61no—
181#127—61no—
182#127—61no—
183C4×Q8C4×D311yescharacteristic
184C4×Q8#6711yescharacteristic
185C4×D4—21no—
186C4×D4—21no—
187C4×D4C4×D311yescharacteristic
188C4×D4#6711yescharacteristic
189#117—21no—
190#117D1211yescharacteristic
191C2×C4⋊C4—61no—
192C2×C42—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8#6711yescharacteristic
203C4⋊C8#6711yescharacteristic
204#104—122no—
205#102—62no—
206#102—62no—
207#102—124no—
208C22⋊C8—22no—
209C22⋊C8—21no—
210#97—22no—
211C4×C8—21no—
212#95—61no—
213C23×C3—21no—
214C23×C3#12111yescharacteristic
215C22×D3—42no—
216C22×D3—82no—
217C22×D3—82no—
218C22×D3—81no—
219C22×D3—81no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—41no—
225C3×D4—41no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—81no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12311yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4#13711yescharacteristic
247C2×Dic3—42no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250D12—44no—
251D12—42no—
252D12—42no—
253D12—82no—
254D12—82no—
255D12—81no—
256D12—81no—
257D12—81no—
258D12—81no—
259C3×C8—42no—
260C3×C8—41no—
261C3×C8—41no—
262C3⋊C8—42no—
263C3⋊C8—81no—
264C24—121no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269C2×Q8(C2×D3)⋊C411yescharacteristic
270C2×D4—21no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—122no—
275C2×D4—122no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—242no—
284C2×D4(C2×D3)⋊C411yescharacteristic
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—121no—
289C22×C4C2×D1211yescharacteristic
290C22×C4#23511yescharacteristic
291C22×C4C2×#6711yescharacteristic
292SD16—124no—
293D8—124no—
294M16—82no—
295M16—121no—
296M16—121no—
297M16—121no—
298M16—121no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—62no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C4⋊C4(C2×D3)⋊C411yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—121no—
318C42—121no—
319C42C2×C4×D311yescharacteristic
320C42#23511yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3#38611yescharacteristic
329C22×C3#46811yescharacteristic
330C22×C3#46911yescharacteristic
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—162no—
335C2×D3—162no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4#41711yescharacteristic
343C3×C4#41811yescharacteristic
344C3×C4#45311yescharacteristic
345C3×C4#45411yescharacteristic
346C3×C4#46811yescharacteristic
347C3×C4#46911yescharacteristic
348Dic3—82no—
349C23—21no—
350C23—122no—
351C23—242no—
352C23—242no—
353C23—241no—
354C23—241no—
355C23#93011yescharacteristic
356Q8—21no—
357Q8—21no—
358D4—21no—
359D4—21no—
360D4—41no—
361D4—41no—
362D4—124no—
363D4—122no—
364D4—122no—
365D4—242no—
366D4—242no—
367D4—241no—
368D4—241no—
369D4—241no—
370D4—241no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—122no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4#90811yescharacteristic
386C2×C4#90811yescharacteristic
387C2×C4#90811yescharacteristic
388C2×C4#90911yescharacteristic
389C2×C4C2×(C2×D3)⋊C411yescharacteristic
390C2×C4C4×#6711yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4#92911yescharacteristic
393C2×C4#93011yescharacteristic
394C2×C4#95311yescharacteristic
395C8—42no—
396C8—41no—
397C8—41no—
398C8—122no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—81no—
403C2×C3#188311yescharacteristic
404C2×C3#189511yescharacteristic
405C2×C3#190311yescharacteristic
406D3—162no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—242no—
413C22—242no—
414C22—242no—
415C22—482no—
416C22—482no—
417C22#513411yescharacteristic
418C22#513511yescharacteristic
419C22#521311yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—242no—
427C4#513111yescharacteristic
428C4#513311yescharacteristic
429C4#513411yescharacteristic
430C4#513511yescharacteristic
431C4#524111yescharacteristic
432C4#524511yescharacteristic
433C3#932211yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—81no—
437C2—482no—
438C2#7166711yescharacteristic
439C2#7199011yescharacteristic
440C2#7202311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2989C2
socleC22×C3#386
prime radical#82526C2
cosocleC2×C3×C42C23
hypercenter#2989D3
hypocenterC3#9322
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2989

Complements

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

NH ≅ G/Ntype
1. C3×#2989C2semidirect
3. #71646C2semidirect
4. #70774C2semidirect
5. #70457C2semidirect
6. #67746C2semidirect
7. #67705C2semidirect
22. #5027C4semidirect
24. #5015C4semidirect
33. #2989D3semidirect
433. C3#9322semidirect

Maximal subgroups

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

subgroupindex
C3×#29892
#825262
#716462
#707742
#704572
#677462
#677052
#93223