Finite Group Databasebeta

← #10779115

Subgroups — #10779115

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83209C211yesmaximal, characteristic
2C2×#5078C211yesmaximal, characteristic, complemented
3#79276C211yesmaximal, characteristic
4#79057C211yesmaximal, characteristic
5C3×#1546C211yesmaximal, characteristic, complemented
6#70778C211yesmaximal, characteristic, complemented
7#70491C211yesmaximal, characteristic, complemented
8#68467C214yesmaximal, complemented
9#68443C214yesmaximal, complemented
10#19271—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#942—21no—
13C2×#909—21no—
14C4×#235C2211yescharacteristic
15C2×#869C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×#423C2211yescharacteristic
18#5225—21no—
19#5120—21no—
20#5090—21no—
21#5090C2212yes—
22#5083—21no—
23#5083C2212yes—
24#5078—21no—
25#5078—21no—
26#5078C2214yes—
27C2×#807—22no—
28#4924C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4849—21no—
31#4849C2212yes—
32#4841—21no—
33#4841C2212yes—
34#4834C2212yes—
35#4830C2212yes—
36#67⋊C8—22no—
37#4685—21no—
38C4×D3×C8—21no—
39C3×#332C2214yes—
40#4486C2214yes—
41#4479C2214yes—
42C2×#500—31no—
43#3039—31no—
44#3039—31no—
45C8×#41—31no—
46#1674—34no—
47#1673—34no—
48#1567—31no—
49#1546—31no—
50#1546D311yescharacteristic, complemented
51C22×C3×Q8—21no—
52C2×#235D411yescharacteristic
53C22×Dic6—21no—
54C2×C3×M16—21no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8C2312yes—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#216—22no—
63#942—44no—
64C2×#215—22no—
65C4×#67—21no—
66C4×#67—21no—
67#926D411yescharacteristic
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×#208—21no—
71C22×C3⋊C8D412yes—
72#909—44no—
73C2×Dic12—22no—
74C2×#204—22no—
75C2×D3×C8—22no—
76C4×D12D411yescharacteristic
77C4×D12C2311yescharacteristic
78#872—21no—
79C42×D3—21no—
80#869—21no—
81#869—21no—
82#869—21no—
83#869C2312yes—
84C4×Dic6D411yescharacteristic
85C4×Dic6C2311yescharacteristic
86C3×C4⋊C8—21no—
87C3×C4⋊C8C2312yes—
88C3×C4⋊C8C2312yes—
89C3×C22⋊C8—21no—
90C3×C4×C8—21no—
91#835—22no—
92#835—22no—
93(C2×D3)⋊C8—22no—
94#816—22no—
95#816—22no—
96Dic3⋊C8—22no—
97C8×Dic3—22no—
98#807—24no—
99#807—24no—
100C4⋊(C3⋊C8)—21no—
101C4⋊(C3⋊C8)C2312yes—
102C4×C3⋊C8D411yescharacteristic
103C4×C3⋊C8D411yescharacteristic
104C4×C3⋊C8C2312yes—
105C2×#137—61no—
106C2×#137—61no—
107C2×C4⋊Q8—31no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C4×#41—31no—
110#500—34no—
111#500—61no—
112#500—61no—
113#462—32no—
114#462—32no—
115#462—61no—
116#462—61no—
117#461—32no—
118#461—32no—
119#461—61no—
120#461—61no—
121C8×Q8—32no—
122C8×D4—32no—
123C8×D4—62no—
124#431—61no—
125#431C2×D311yescharacteristic
126C2×#107—62no—
127#423—31no—
128#423C2×D311yescharacteristic
129#419—61no—
130#419—61no—
131C2×C4×C8—31no—
132C2×C4×C8—61no—
133#335—34no—
134#332—34no—
135#332C2×D314yes—
136C2×C3×Q8—22no—
137C2×C3×Q8—41no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C22×C3×C4—41no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C2×#67—21no—
145C22×Dic3—41no—
146#235—22no—
147#235D814yes—
148C2×D12C2×D411yescharacteristic
149C2×C4×D3—21no—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×Dic6—22no—
153C2×Dic6—42no—
154C2×Dic6—42no—
155C2×Dic6—41no—
156C2×Dic6C2×D411yescharacteristic
157C3×M16—42no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167C3×C42C2411yescharacteristic
168#217—21no—
169#216—44no—
170#215—44no—
171(C2×D3)⋊C4—21no—
172(C2×D3)⋊C4—21no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—41no—
176C4⋊Dic3—41no—
177C4⋊Dic3C2×D411yescharacteristic
178C4⋊Dic3C2×D411yescharacteristic
179Dic3⋊C4—21no—
180Dic3⋊C4—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184#208—42no—
185C2×C3⋊C8—24no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8C2×D412yes—
188C2×C3⋊C8C2×D412yes—
189Dic12—44no—
190#204—44no—
191D3×C8—44no—
192C2×#41—31no—
193C22×Q8—21no—
194C22×Q8—61no—
195#137—124no—
196#137—124no—
197C2×Q16—62no—
198C2×Q16—62no—
199C2×SD16—62no—
200C2×SD16—62no—
201C2×M16—21no—
202C2×M16—61no—
203C22×C8—32no—
204C22×C8—62no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—32no—
209C4⋊Q8—61no—
210C4⋊Q8—61no—
211C4⋊Q8—61no—
212C4⋊Q8C22×D312yes—
213C4×Q8—31no—
214C4×Q8—31no—
215C4×D4—31no—
216C4×D4—31no—
217C4×D4—61no—
218C4×D4—61no—
219#117—31no—
220#117—61no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227#107—64no—
228#107—64no—
229C4⋊C8—21no—
230C4⋊C8—32no—
231C4⋊C8—62no—
232C4⋊C8—61no—
233C4⋊C8C22×D312yes—
234C4⋊C8C22×D312yes—
235#103—62no—
236#103—62no—
237#103—62no—
238#103—62no—
239C22⋊C8—21no—
240C22⋊C8—62no—
241C4×C8—21no—
242C4×C8—31no—
243C4×C8—31no—
244C4×C8—32no—
245C4×C8—62no—
246C23×C3#12111yescharacteristic
247C22×D3—21no—
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8—82no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—42no—
254C2×C3×C4—82no—
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4C2×D812yes—
259C2×C3×C4#13512yes—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263#67—24no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—42no—
268C2×Dic3—82no—
269D12—21no—
270D12C2×D812yes—
271C4×D3—24no—
272C4×D3—42no—
273C4×D3—42no—
274Dic6—21no—
275Dic6—42no—
276Dic6—42no—
277Dic6—82no—
278Dic6C2×D812yes—
279C3×C8—42no—
280C3×C8—42no—
281C3⋊C8—22no—
282C3⋊C8—42no—
283C3⋊C8#13714yescomplemented
284#41—34no—
285#41—62no—
286C2×Q8—22no—
287C2×Q8—31no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—62no—
292C2×Q8—121no—
293C2×Q8—122no—
294C2×Q8—122no—
295C2×D4—31no—
296C2×D4—61no—
297C22×C4—41no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—121no—
302C22×C4#23511yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305Q16—124no—
306Q16—124no—
307SD16—124no—
308SD16—124no—
309M16—42no—
310M16—122no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—32no—
314C2×C8—32no—
315C2×C8—64no—
316C2×C8—62no—
317C2×C8—124no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—31no—
321C4⋊C4—31no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—62no—
327C4⋊C4—62no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C42—61no—
334C42—61no—
335C42—61no—
336C42#23511yescharacteristic
337C42D3×D411yescharacteristic
338C42D3×D411yescharacteristic
339C42C23×D311yescharacteristic
340C22×C3—21no—
341C22×C3—21no—
342C22×C3#46612yes—
343C22×C3C2×#12111yescharacteristic
344C2×D3—22no—
345C2×D3—41no—
346C2×D3—41no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—82no—
350C3×C4#46612yes—
351C3×C4#47312yes—
352C3×C4C22×D811yescharacteristic
353C3×C4C2×#13511yescharacteristic
354C3×C4C2×#13712yes—
355Dic3—22no—
356Dic3—41no—
357Dic3—41no—
358Dic3—82no—
359C23—61no—
360C23#88411yescharacteristic
361Q8—32no—
362Q8—42no—
363Q8—42no—
364Q8—61no—
365Q8—82no—
366Q8—122no—
367Q8—122no—
368Q8—242no—
369D4—32no—
370D4—64no—
371D4—61no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—61no—
376C2×C4—64no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—82no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—242no—
384C2×C4#88411yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4D3×D812yes—
388C2×C4#91612yes—
389C2×C4C2×#23511yescharacteristic
390C2×C4C2×D3×D411yescharacteristic
391C2×C4C2×D3×D411yescharacteristic
392C8—34no—
393C8—42no—
394C8—42no—
395C8—62no—
396C8—122no—
397C2×C3—22no—
398C2×C3#303911yescharacteristic
399C2×C3C2×#46611yescharacteristic
400C2×C3C2×#47311yescharacteristic
401D3—42no—
402C22—21no—
403C22—21no—
404C22—62no—
405C22—121no—
406C22—121no—
407C22#477512yes—
408C22C2×#88411yescharacteristic
409C4—21no—
410C4—21no—
411C4—62no—
412C4—82no—
413C4—121no—
414C4—121no—
415C4—242no—
416C4#475512yes—
417C4#477512yes—
418C4C2×D3×D811yescharacteristic
419C4C2×#91611yescharacteristic
420C4D3×#13712yes—
421C3#1927111yescharacteristic, complemented, Sylow
422C2—22no—
423C2—122no—
424C2C2×#475511yescharacteristic
425C2C2×#477511yescharacteristic
426C2#7897411yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1546C2
socleC22×C3C2×#121
prime radicalC2×#235D4
hypercenter#1546D3
hypocenterC3#19271
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
2. C2×#5078C2semidirect
5. C3×#1546C2semidirect
6. #70778C2semidirect
7. #70491C2semidirect
8. #68467C2semidirect
9. #68443C2semidirect
50. #1546D3semidirect
283. C3⋊C8#137semidirect
421. C3#19271semidirect

Maximal subgroups

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

subgroupindex
#832092
C2×#50782
#792762
#790572
C3×#15462
#707782
#704912
#684672
#684432
#192713