Finite Group Databasebeta

← #10694581

Subgroups — #10694581

Order 768.

478 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#83285C211yesmaximal, characteristic
2C3×#1517C211yesmaximal, characteristic, complemented
3#71814C211yesmaximal, characteristic, complemented
4#70790C211yesmaximal, characteristic, complemented
5#70742C211yesmaximal, characteristic, complemented
6#69547C211yesmaximal, characteristic, complemented
7#69545C211yesmaximal, characteristic, complemented
8#9760—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#950—21no—
11C2×#950—21no—
12#5477C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5092C411yescharacteristic, complemented
16#5085C411yescharacteristic, complemented
17#5076C411yescharacteristic, complemented
18#5073C411yescharacteristic, complemented
19#5059—21no—
20#5059—21no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#327—22no—
25#4537—22no—
26#4529—21no—
27#4529—21no—
28#4470—22no—
29#4455—21no—
30#4454—21no—
31#4451—22no—
32#3147—31no—
33#1835—31no—
34#1517—31no—
35#1517—31no—
36#1517—31no—
37#1517D311yescharacteristic, complemented
38#1452—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×C42C2311yescharacteristic
50#950—42no—
51#950—42no—
52#950—42no—
53#950—42no—
54C2×#216—21no—
55C2×#216—21no—
56C2×#215—21no—
57C2×#215—21no—
58C2×#214—21no—
59C2×#214—21no—
60C2×#213—21no—
61C2×#213—21no—
62#930—21no—
63#930—21no—
64C2×C4⋊Dic3—21no—
65C22×C3⋊C8D411yescharacteristic
66C22×C3⋊C8D411yescharacteristic
67#875—21no—
68#874C2×C411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869C2×C411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C22⋊C8D412yes—
74C3×C4×C8—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830D412yes—
80C4⋊(C3⋊C8)—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8C2×C411yescharacteristic
84C4×C3⋊C8D411yescharacteristic
85C4×C3⋊C8D411yescharacteristic
86C2×#135—61no—
87C2×#135—61no—
88#532—31no—
89#532C2×D311yescharacteristic
90#495—61no—
91#495—61no—
92#494—31no—
93#493—31no—
94#492—31no—
95#492—31no—
96#431—31no—
97#431C2×D311yescharacteristic
98C2×C4×C8—31no—
99#382—31no—
100#382C2×D311yescharacteristic
101#340—61no—
102#340—61no—
103#327—22no—
104#327—62no—
105#327—62no—
106#327—62no—
107#322—61no—
108#322—61no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C2×C3×Q8—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—41no—
124C22×Dic3—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—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145#216—42no—
146#216—42no—
147#215—42no—
148#215—42no—
149#214—42no—
150#214—42no—
151#213—42no—
152#213—42no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8C22⋊C411yescharacteristic
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8C22⋊C411yescharacteristic
167C2×C3⋊C8C22⋊C411yescharacteristic
168C2×#41—21no—
169C2×#41—61no—
170#135—122no—
171#135—122no—
172#135—122no—
173#135—122no—
174C2×Q16—61no—
175C2×Q16—61no—
176C2×SD16—61no—
177C2×SD16—61no—
178C2×SD16—61no—
179C2×SD16—61no—
180C2×D8—61no—
181C2×D8—61no—
182C22×C8—31no—
183C22×C8—31no—
184C4⋊Q8—31no—
185C4⋊Q8C4×D311yescharacteristic
186#127—31no—
187#127C4×D311yescharacteristic
188#124—21no—
189#124—61no—
190#121—21no—
191#121—21no—
192#121—61no—
193#121—61no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—61no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—61no—
203C22⋊C8—32no—
204C22⋊C8#6712yes—
205C4×C8—21no—
206C4×C8—31no—
207C4×C8—31no—
208C4×C8—31no—
209C4×C8—31no—
210C4×C8—61no—
211#95—21no—
212#95—21no—
213#95—61no—
214#95—61no—
215C23×C3—41no—
216C23×C3#12711yescharacteristic
217C22×D3—41no—
218C3×Q8—41no—
219C3×Q8—41no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4C2×C22⋊C411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12411yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244#67—81no—
245#67—81no—
246C2×Dic3—41no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250D12—41no—
251D12—41no—
252D12—81no—
253D12—81no—
254C4×D3—81no—
255C4×D3—81no—
256Dic6—41no—
257Dic6—41no—
258C3×C8—42no—
259C3⋊C8—21no—
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C3⋊C8—42no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—121no—
269#41—121no—
270#41—121no—
271#41—121no—
272C2×Q8—21no—
273C2×Q8—61no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—121no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—121no—
285C22×C4—121no—
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4C2×#6711yescharacteristic
289Q16—122no—
290Q16—122no—
291SD16—122no—
292SD16—122no—
293SD16—122no—
294SD16—122no—
295D8—122no—
296D8—122no—
297C2×C8—22no—
298C2×C8—31no—
299C2×C8—31no—
300C2×C8—31no—
301C2×C8—31no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—62no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42C2×C4×D311yescharacteristic
320C42#23511yescharacteristic
321C42D3×D411yescharacteristic
322C42#23711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#39011yescharacteristic
329C22×C3#49611yescharacteristic
330C22×C3#49611yescharacteristic
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—81no—
339C3×C4—161no—
340C3×C4#41811yescharacteristic
341C3×C4#41811yescharacteristic
342C3×C4#49011yescharacteristic
343C3×C4#49011yescharacteristic
344Dic3—81no—
345Dic3—161no—
346C23—41no—
347C23—121no—
348C23#94011yescharacteristic
349Q8—41no—
350Q8—41no—
351Q8—121no—
352Q8—121no—
353D4—41no—
354D4—41no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—121no—
360D4—121no—
361D4—241no—
362D4—241no—
363D4—241no—
364D4—241no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4D3×C22⋊C411yescharacteristic
380C2×C4#67⋊C411yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#88511yescharacteristic
383C2×C4#91111yescharacteristic
384C2×C4#91111yescharacteristic
385C2×C4#91411yescharacteristic
386C2×C4#91411yescharacteristic
387C2×C4C4×#6711yescharacteristic
388C2×C4#93911yescharacteristic
389C8—42no—
390C8—61no—
391C8—61no—
392C8—61no—
393C8—61no—
394C8—122no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—161no—
398C2×C3#196111yescharacteristic
399C2×C3#197611yescharacteristic
400C2×C3#197611yescharacteristic
401D3—161no—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22#495711yescharacteristic
411C22#515411yescharacteristic
412C22#517611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—161no—
419C4—241no—
420C4—481no—
421C4#476411yescharacteristic
422C4#476411yescharacteristic
423C4#477011yescharacteristic
424C4#477011yescharacteristic
425C3#976011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—481no—
430C2#7004211yescharacteristic
431C2#7016011yescharacteristic
432C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#390
prime radical#83285C2
cosocleC2×C3×C42C23
hypercenter#1517D3
hypocenterC3#9760
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. C3×#1517C2semidirect
3. #71814C2semidirect
4. #70790C2semidirect
5. #70742C2semidirect
6. #69547C2semidirect
7. #69545C2semidirect
15. #5092C4semidirect
16. #5085C4semidirect
17. #5076C4semidirect
18. #5073C4semidirect
37. #1517D3semidirect
425. C3#9760semidirect

Maximal subgroups

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

subgroupindex
#832852
C3×#15172
#718142
#707902
#707422
#695472
#695452
#97603