Finite Group Databasebeta

← #10694547

Subgroups — #10694547

Order 768.

502 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83285C211yesmaximal, characteristic
2C3×#1502C211yesmaximal, characteristic, complemented
3#71800C211yesmaximal, characteristic, complemented
4#70758C211yesmaximal, characteristic, complemented
5#70745C211yesmaximal, characteristic, complemented
6#69477C211yesmaximal, characteristic, complemented
7#69469C211yesmaximal, characteristic, complemented
8#9753—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#950—21no—
11C2×#950—21no—
12#5477C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5092C411yescharacteristic, complemented
16#5085C411yescharacteristic, complemented
17#5076C411yescharacteristic, complemented
18#5073C411yescharacteristic, complemented
19#5059—21no—
20#5059—21no—
21#4925C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×#803—21no—
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×#327—22no—
26D4⋊(C3⋊C8)—22no—
27#4524—21no—
28#4480—21no—
29#4479—21no—
30#4459—22no—
31#4451—22no—
32#3147—31no—
33#1770—31no—
34#1564—31no—
35#1502—31no—
36#1502D311yescharacteristic, complemented
37#1480—31no—
38#1480—31no—
39C2×C3×#41—21no—
40C2×#235—21no—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#127C2×C411yescharacteristic
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51#950—42no—
52#950—42no—
53#950—42no—
54#950—42no—
55C2×#216—21no—
56C2×#216—21no—
57C2×#215—21no—
58C2×#215—21no—
59C2×#214—21no—
60C2×#214—21no—
61C2×#213—21no—
62C2×#213—21no—
63#930—21no—
64#930—21no—
65C22×C3⋊C8—21no—
66C22×C3⋊C8D411yescharacteristic
67C22×C3⋊C8D411yescharacteristic
68#875—21no—
69#874C2×C411yescharacteristic
70C4×D12—21no—
71C4×D12—21no—
72#869C2×C411yescharacteristic
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3×#95D412yes—
77#830—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—42no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8C2×C411yescharacteristic
84C4×C3⋊C8C2×C411yescharacteristic
85C4×C3⋊C8D411yescharacteristic
86C4×C3⋊C8D411yescharacteristic
87C2×#135—61no—
88C2×#135—61no—
89#532—31no—
90#532C2×D311yescharacteristic
91#495—61no—
92#495—61no—
93#494—31no—
94#493—31no—
95#492—31no—
96#492—31no—
97#431—31no—
98#431C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100C2×#97—61no—
101C2×C4×C8—31no—
102C4×C4⋊C4C2×D311yescharacteristic
103#336—61no—
104#335—61no—
105#334—61no—
106#327—22no—
107#327—62no—
108D4⋊C8—62no—
109D4⋊C8—62no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C2×#67—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—42no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#216—42no—
147#216—42no—
148#215—42no—
149#215—42no—
150#214—42no—
151#214—42no—
152#213—42no—
153#213—42no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8C22⋊C411yescharacteristic
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8C22⋊C411yescharacteristic
167C2×C3⋊C8C22⋊C411yescharacteristic
168C2×C3⋊C8D812yes—
169C2×C3⋊C8SD1612yes—
170C2×#41—21no—
171C2×#41—61no—
172#135—122no—
173#135—122no—
174#135—122no—
175#135—122no—
176C2×Q16—61no—
177C2×Q16—61no—
178C2×SD16—61no—
179C2×SD16—61no—
180C2×SD16—61no—
181C2×SD16—61no—
182C2×D8—61no—
183C2×D8—61no—
184C22×C8—31no—
185C22×C8—31no—
186C22×C8—61no—
187C4⋊Q8—31no—
188C4⋊Q8C4×D311yescharacteristic
189#127—31no—
190#127C4×D311yescharacteristic
191#124—21no—
192#124—61no—
193#121—21no—
194#121—21no—
195#121—61no—
196#121—61no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—61no—
200C4×D4—61no—
201C2×C4⋊C4—21no—
202C2×C42—21no—
203C2×C42C22×D311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210#97—122no—
211C4×C8—21no—
212C4×C8—31no—
213C4×C8—31no—
214C4×C8—31no—
215C4×C8—31no—
216C4×C8—61no—
217#95#6712yes—
218C23×C3—41no—
219C23×C3#12111yescharacteristic
220C22×D3—41no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—42no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12411yescharacteristic
242C2×C3×C4#12711yescharacteristic
243C2×C3×C4C2×D811yescharacteristic
244C2×C3×C4C2×SD1611yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247#67—81no—
248#67—81no—
249C2×Dic3—41no—
250D12—41no—
251D12—41no—
252D12—42no—
253D12—81no—
254C4×D3—81no—
255C4×D3—81no—
256Dic6—41no—
257Dic6—41no—
258C3×C8—81no—
259C3⋊C8—21no—
260C3⋊C8—21no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263C3⋊C8#10212yes—
264C3⋊C8#10212yes—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—121no—
270#41—121no—
271#41—121no—
272#41—121no—
273C2×Q8—21no—
274C2×Q8—61no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—121no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—121no—
286C22×C4D3×D411yescharacteristic
287C22×C4#23711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289Q16—122no—
290Q16—122no—
291SD16—122no—
292SD16—122no—
293SD16—122no—
294SD16—122no—
295D8—122no—
296D8—122no—
297C2×C8—31no—
298C2×C8—31no—
299C2×C8—32no—
300C2×C8—32no—
301C2×C8—31no—
302C2×C8—31no—
303C2×C8—41no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—62no—
307C2×C8—121no—
308C2×C8—121no—
309C2×C8—121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—42no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42—41no—
320C42—41no—
321C42C2×C4×D311yescharacteristic
322C42#23511yescharacteristic
323C42D3×D411yescharacteristic
324C42D3×D411yescharacteristic
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#39011yescharacteristic
331C22×C3#45911yescharacteristic
332C22×C3#46011yescharacteristic
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4—81no—
341C3×C4C2×#10211yescharacteristic
342C3×C4#41811yescharacteristic
343C3×C4#44412yes—
344C3×C4#45911yescharacteristic
345C3×C4#46011yescharacteristic
346Dic3—81no—
347C23—41no—
348C23—121no—
349C23#93911yescharacteristic
350Q8—41no—
351Q8—41no—
352Q8—121no—
353Q8—121no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—121no—
361D4—121no—
362D4—122no—
363D4—241no—
364D4—241no—
365D4—241no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—121no—
375C2×C4—241no—
376C2×C4—241no—
377C2×C4D3×C22⋊C411yescharacteristic
378C2×C4#67⋊C411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4#88511yescharacteristic
381C2×C4D3×D811yescharacteristic
382C2×C4#91111yescharacteristic
383C2×C4D3×SD1611yescharacteristic
384C2×C4#91411yescharacteristic
385C2×C4C4×#6711yescharacteristic
386C2×C4#94011yescharacteristic
387C8—32no—
388C8—32no—
389C8—61no—
390C8—61no—
391C8—81no—
392C8—122no—
393C8—241no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—161no—
397C2×C3#197511yescharacteristic
398C2×C3#197611yescharacteristic
399C2×C3#198211yescharacteristic
400D3—161no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22#495711yescharacteristic
410C22#515011yescharacteristic
411C22#516311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—82no—
416C4—81no—
417C4—241no—
418C4#476211yescharacteristic
419C4D3×#10211yescharacteristic
420C4#477011yescharacteristic
421C4#478611yescharacteristic
422C4#482312yes—
423C3#975311yescharacteristic, complemented, Sylow
424C2—21no—
425C2—21no—
426C2—161no—
427C2—481no—
428C2#7004211yescharacteristic
429C2#7016111yescharacteristic
430C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#390
prime radical#83285C2
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9753
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. C3×#1502C2semidirect
3. #71800C2semidirect
4. #70758C2semidirect
5. #70745C2semidirect
6. #69477C2semidirect
7. #69469C2semidirect
15. #5092C4semidirect
16. #5085C4semidirect
17. #5076C4semidirect
18. #5073C4semidirect
36. #1502D3semidirect
423. C3#9753semidirect

Maximal subgroups

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

subgroupindex
#832852
C3×#15022
#718002
#707582
#707452
#694772
#694692
#97533