Finite Group Databasebeta

← #10694563

Subgroups — #10694563

Order 768.

502 conjugacy classes of subgroups.

The subgroup lattice has 438 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#70740C211yesmaximal, characteristic, complemented
6#69549C211yesmaximal, characteristic, complemented
7#69545C211yesmaximal, characteristic, complemented
8#9756—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—
21C2×#808—21no—
22C2×#807—21no—
23#4936C2211yescharacteristic
24#4925C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C3×#327—22no—
27#4537—22no—
28#4470—22no—
29#4454—21no—
30C8×C3⋊C8—21no—
31#4451—22no—
32#3147—31no—
33#1846—31no—
34#1517—31no—
35#1517—31no—
36#1517—31no—
37#1517D311yescharacteristic, complemented
38#1450—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—
80#808—42no—
81#807—42no—
82C4⋊(C3⋊C8)—21no—
83C4×C3⋊C8—21no—
84C4×C3⋊C8C2×C411yescharacteristic
85C4×C3⋊C8C2×C411yescharacteristic
86C4×C3⋊C8D411yescharacteristic
87C4×C3⋊C8D411yescharacteristic
88C2×#135—61no—
89C2×#135—61no—
90#532—31no—
91#532C2×D311yescharacteristic
92#495—61no—
93#495—61no—
94#494—31no—
95#493—31no—
96#492—31no—
97#492—31no—
98#431—31no—
99#431C2×D311yescharacteristic
100C2×#107—61no—
101C2×#106—61no—
102C2×C4×C8—31no—
103#382—31no—
104#382C2×D311yescharacteristic
105#327—22no—
106#327—62no—
107#327—62no—
108#327—62no—
109#322—61no—
110C82—61no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C2×C3×Q8—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×#67—41no—
126C22×Dic3—41no—
127#235—41no—
128#235—41no—
129#235—41no—
130#235—41no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×Dic6—21no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147#216—42no—
148#216—42no—
149#215—42no—
150#215—42no—
151#214—42no—
152#214—42no—
153#213—42no—
154#213—42no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8C22⋊C411yescharacteristic
163C2×C3⋊C8C22⋊C411yescharacteristic
164C2×C3⋊C8C22⋊C411yescharacteristic
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8D812yes—
167C2×C3⋊C8D812yes—
168C2×C3⋊C8SD1612yes—
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—
186C4⋊Q8—31no—
187C4⋊Q8C4×D311yescharacteristic
188#127—31no—
189#127C4×D311yescharacteristic
190#124—21no—
191#124—61no—
192#121—21no—
193#121—21no—
194#121—61no—
195#121—61no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—61no—
199C4×D4—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—61no—
202C2×C42C22×D311yescharacteristic
203#107—122no—
204#106—122no—
205C4⋊C8—21no—
206C4⋊C8—61no—
207C22⋊C8—32no—
208C22⋊C8#6712yes—
209C4×C8—21no—
210C4×C8—31no—
211C4×C8—31no—
212C4×C8—31no—
213C4×C8—31no—
214C4×C8—61no—
215#95—21no—
216#95—21no—
217#95—61no—
218#95—61no—
219C23×C3—41no—
220C23×C3#12711yescharacteristic
221C22×D3—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—81no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12411yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4C2×SD1611yescharacteristic
247C2×C3×C4C2×SD1611yescharacteristic
248#67—81no—
249#67—81no—
250C2×Dic3—41no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254D12—41no—
255D12—41no—
256D12—81no—
257D12—81no—
258C4×D3—81no—
259C4×D3—81no—
260Dic6—41no—
261Dic6—41no—
262C3×C8—42no—
263C3⋊C8—42no—
264C3⋊C8#10212yes—
265C3⋊C8#10212yes—
266C3⋊C8#10212yes—
267C3⋊C8#10212yes—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—121no—
273#41—121no—
274#41—121no—
275#41—121no—
276C2×Q8—21no—
277C2×Q8—61no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—121no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—121no—
289C22×C4—121no—
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4C2×#6711yescharacteristic
293Q16—122no—
294Q16—122no—
295SD16—122no—
296SD16—122no—
297SD16—122no—
298SD16—122no—
299D8—122no—
300D8—122no—
301C2×C8—22no—
302C2×C8—31no—
303C2×C8—31no—
304C2×C8—31no—
305C2×C8—31no—
306C2×C8—32no—
307C2×C8—32no—
308C2×C8—32no—
309C2×C8—32no—
310C2×C8—62no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C42C2×C4×D311yescharacteristic
324C42#23511yescharacteristic
325C42D3×D411yescharacteristic
326C42#23711yescharacteristic
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3#39011yescharacteristic
333C22×C3#49211yescharacteristic
334C22×C3#49311yescharacteristic
335C2×D3—81no—
336C2×D3—81no—
337C2×D3—81no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—81no—
343C3×C4—161no—
344C3×C4C2×#10211yescharacteristic
345C3×C4C2×#10211yescharacteristic
346C3×C4#48611yescharacteristic
347C3×C4#48611yescharacteristic
348Dic3—81no—
349Dic3—161no—
350C23—41no—
351C23—121no—
352C23#94011yescharacteristic
353Q8—41no—
354Q8—41no—
355Q8—121no—
356Q8—121no—
357D4—41no—
358D4—41no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—121no—
364D4—121no—
365D4—241no—
366D4—241no—
367D4—241no—
368D4—241no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—121no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4D3×C22⋊C411yescharacteristic
384C2×C4#67⋊C411yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#88511yescharacteristic
387C2×C4D3×D811yescharacteristic
388C2×C4D3×D811yescharacteristic
389C2×C4D3×SD1611yescharacteristic
390C2×C4D3×SD1611yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4#93911yescharacteristic
393C8—32no—
394C8—32no—
395C8—32no—
396C8—32no—
397C8—42no—
398C8—122no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—161no—
402C2×C3#195711yescharacteristic
403C2×C3#197611yescharacteristic
404C2×C3#197611yescharacteristic
405D3—161no—
406C22—21no—
407C22—21no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—241no—
412C22—241no—
413C22—241no—
414C22#495711yescharacteristic
415C22#515111yescharacteristic
416C22#517511yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—161no—
423C4—241no—
424C4—481no—
425C4#476011yescharacteristic
426C4#476011yescharacteristic
427C4D3×#10211yescharacteristic
428C4D3×#10211yescharacteristic
429C3#975611yescharacteristic, complemented, Sylow
430C2—21no—
431C2—21no—
432C2—161no—
433C2—481no—
434C2#7004211yescharacteristic
435C2#7015711yescharacteristic
436C2#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#9756
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. #70740C2semidirect
6. #69549C2semidirect
7. #69545C2semidirect
15. #5092C4semidirect
16. #5085C4semidirect
17. #5076C4semidirect
18. #5073C4semidirect
37. #1517D3semidirect
429. C3#9756semidirect

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
#707402
#695492
#695452
#97563