Finite Group Databasebeta

← #10694345

Subgroups — #10694345

Order 768.

506 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83291C211yesmaximal, characteristic
2C3×#1502C211yesmaximal, characteristic, complemented
3#71818C211yesmaximal, characteristic, complemented
4#70789C211yesmaximal, characteristic, complemented
5#70739C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic, complemented
7C4×#807C211yesmaximal, characteristic, complemented
8#9719—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13C2×#837—22no—
14#5088C411yescharacteristic, complemented
15#5078C411yescharacteristic, complemented
16#5073C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5038—21no—
19#5038—21no—
20C4×#213—21no—
21C4×#213—21no—
22C2×#807—21no—
23C4×C4⋊Dic3C2211yescharacteristic
24C2×#805—22no—
25C4×#208C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27C3×#327—22no—
28#4529—21no—
29#4470—22no—
30#4454—21no—
31C8×C3⋊C8—21no—
32#3169—31no—
33#1980—31no—
34#1980—31no—
35C4×#107—31no—
36#1502—31no—
37#1502D311yescharacteristic, complemented
38C8×M16—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
51C2×#213—21no—
52C2×#213—21no—
53#930—21no—
54#930—21no—
55C2×C4⋊Dic3—21no—
56C2×C4×Dic3—21no—
57C2×#208D412yes—
58C22×C3⋊C8D411yescharacteristic
59C22×C3⋊C8D411yescharacteristic
60#875—21no—
61#874C2×C411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869C2×C411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C22⋊C8D412yes—
67C3×C4×C8—21no—
68C3×#95—21no—
69#837—44no—
70#832—21no—
71#832—21no—
72C3⋊#95—21no—
73#809—21no—
74#809—21no—
75#807—24no—
76#807—21no—
77#807—21no—
78#805—44no—
79#803—21no—
80C4×C3⋊C8—21no—
81C4×C3⋊C8C2×C411yescharacteristic
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8D411yescharacteristic
84C4×C3⋊C8D411yescharacteristic
85#532—31no—
86#532C2×D311yescharacteristic
87#500—31no—
88#493—31no—
89#486—31no—
90#486—31no—
91#466—61no—
92#466—61no—
93C4×D8—61no—
94C4×D8—61no—
95#431C2×D311yescharacteristic
96C2×#107—61no—
97C2×#104—62no—
98C2×#104—62no—
99C4×M16—31no—
100C2×C4×C8—31no—
101C4×C4⋊C4—31no—
102C4×C4⋊C4C2×D311yescharacteristic
103#340—61no—
104#327—22no—
105#327—62no—
106#322—61no—
107C82—61no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—41no—
121C22×Dic3—41no—
122#235—42no—
123#235—42no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C2×C3×C8—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—42no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141#213—42no—
142#213—42no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—41no—
149C4×Dic3—42no—
150#208—22no—
151#208D814yescomplemented
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8C22⋊C411yescharacteristic
156C2×C3⋊C8C22⋊C411yescharacteristic
157C2×C3⋊C8C22⋊C411yescharacteristic
158C2×C3⋊C8C22⋊C411yescharacteristic
159C2×#41—21no—
160C2×#41—61no—
161C2×D8—61no—
162C2×D8—61no—
163C2×M16—32no—
164C22×C8—31no—
165C22×C8—31no—
166C4⋊Q8—31no—
167C4⋊Q8C4×D311yescharacteristic
168#127—31no—
169#127C4×D311yescharacteristic
170#124—21no—
171#124—61no—
172#121—21no—
173#121—21no—
174#121—61no—
175#121—61no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—61no—
179C4×D4—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—61no—
182C2×C42—21no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#107—61no—
186#107—61no—
187#107—64no—
188C4⋊C8—21no—
189#104—124no—
190#104—124no—
191#102—61no—
192#102—61no—
193#102—61no—
194#102—61no—
195C22⋊C8#6712yes—
196#97—61no—
197C4×C8—21no—
198C4×C8—31no—
199C4×C8—31no—
200C4×C8—31no—
201C4×C8—31no—
202C4×C8—61no—
203#95—21no—
204#95—61no—
205C23×C3—41no—
206C23×C3#12711yescharacteristic
207C22×D3—41no—
208C3×Q8—42no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—42no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4—82no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12411yescharacteristic
229C2×C3×C4C2×D812yes—
230C2×C3×C4#13512yes—
231#67—81no—
232#67—81no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237D12—41no—
238D12—41no—
239D12—42no—
240D12—81no—
241C4×D3—82no—
242Dic6—42no—
243C3×C8—42no—
244C3⋊C8—21no—
245C3⋊C8—21no—
246C3⋊C8—21no—
247C3⋊C8—24no—
248C3⋊C8#10412yescomplemented
249#41—42no—
250#41—42no—
251#41—122no—
252#41—122no—
253C2×Q8—21no—
254C2×Q8—61no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—121no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4C2×#6711yescharacteristic
270D8—122no—
271D8—122no—
272M16—34no—
273M16—62no—
274C2×C8—22no—
275C2×C8—31no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—121no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—62no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—42no—
295C42—122no—
296C42C2×C4×D311yescharacteristic
297C42#23511yescharacteristic
298C42D3×D411yescharacteristic
299C42#23711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3#39011yescharacteristic
306C22×C3#49311yescharacteristic
307C22×C3#49511yescharacteristic
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4—81no—
316C3×C4C2×#10411yescharacteristic
317C3×C4#42111yescharacteristic
318C3×C4C4×D812yes—
319C3×C4#46612yes—
320Dic3—82no—
321Dic3—81no—
322C23—41no—
323C23—121no—
324C23#94011yescharacteristic
325Q8—42no—
326Q8—122no—
327D4—42no—
328D4—41no—
329D4—41no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—122no—
334D4—121no—
335D4—121no—
336D4—241no—
337D4—241no—
338D4—241no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—241no—
350C2×C4—241no—
351C2×C4—242no—
352C2×C4D3×C22⋊C411yescharacteristic
353C2×C4#67⋊C411yescharacteristic
354C2×C4#88411yescharacteristic
355C2×C4#88511yescharacteristic
356C2×C4D3×D812yes—
357C2×C4#91612yes—
358C2×C4C4×#6711yescharacteristic
359C2×C4#93911yescharacteristic
360C8—32no—
361C8—42no—
362C8—64no—
363C8—61no—
364C8—61no—
365C8—61no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—161no—
369C2×C3#197011yescharacteristic
370C2×C3#197611yescharacteristic
371C2×C3#198011yescharacteristic
372D3—161no—
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22#495711yescharacteristic
382C22#515111yescharacteristic
383C22#516811yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—82no—
388C4—81no—
389C4—242no—
390C4—241no—
391C4#475612yes—
392C4#477512yes—
393C4D3×#10411yescharacteristic
394C4#482111yescharacteristic
395C3#971911yescharacteristic, complemented, Sylow
396C2—21no—
397C2—21no—
398C2—161no—
399C2—481no—
400C2#7004211yescharacteristic
401C2#7019011yescharacteristic
402C2#7035811yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#390
prime radical#83291C2
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9719
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. #71818C2semidirect
4. #70789C2semidirect
5. #70739C2semidirect
6. #69492C2semidirect
7. C4×#807C2semidirect
14. #5088C4semidirect
15. #5078C4semidirect
16. #5073C4semidirect
17. #5070C4semidirect
37. #1502D3semidirect
151. #208D8semidirect
248. C3⋊C8#104semidirect
395. C3#9719semidirect

Maximal subgroups

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

subgroupindex
#832912
C3×#15022
#718182
#707892
#707392
#694922
C4×#8072
#97193