Finite Group Databasebeta

← #10950905

Subgroups — #10950905

Order 768.

548 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79725C211yesmaximal, characteristic, complemented
2#79598C211yesmaximal, characteristic
3C3×#2175C211yesmaximal, characteristic, complemented
4#71488C211yesmaximal, characteristic, complemented
5#71479C211yesmaximal, characteristic, complemented
6#70913C211yesmaximal, characteristic, complemented
7#70884C211yesmaximal, characteristic, complemented
8#69329C211yesmaximal, characteristic, complemented
9#69320C211yesmaximal, characteristic, complemented
10#69230C211yesmaximal, characteristic, complemented
11#69215C211yesmaximal, characteristic, complemented
12#69198C211yesmaximal, characteristic, complemented
13#69183C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15#69071C211yesmaximal, characteristic, complemented
16#33751—31nomaximal, Sylow
17#5578—21no—
18#5548—21no—
19C2×C3×SD32C2211yescharacteristic
20C3×C4×Q16C2211yescharacteristic
21C3×C4×D8C2211yescharacteristic
22#5188C2211yescharacteristic
23#5183C2211yescharacteristic
24C2×#828C2211yescharacteristic
25#5157C2211yescharacteristic
26#5151C2211yescharacteristic
27C2×#827C2211yescharacteristic
28C4×#216—21no—
29C4×#213—21no—
30C2×#800C2211yescharacteristic
31C2×#798C2211yescharacteristic
32C2×#798C2211yescharacteristic
33#4887C2211yescharacteristic
34#4883C2211yescharacteristic
35#4874C2211yescharacteristic
36#4874C2211yescharacteristic
37#4813—21no—
38#4792—21no—
39#4775—21no—
40#4760—21no—
41C4×Dic12C2211yescharacteristic
42C4×D24C2211yescharacteristic
43C4×D3×C8C2211yescharacteristic
44C3×#366C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#357C2211yescharacteristic
47C3×#346C2211yescharacteristic
48#4564C2211yescharacteristic
49#4562C2211yescharacteristic
50#4509C2211yescharacteristic
51#4504C2211yescharacteristic
52#4503C2211yescharacteristic
53Dic3×C16C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55#4492C2211yescharacteristic
56#4491C2211yescharacteristic
57#4489C2211yescharacteristic
58#4489C2211yescharacteristic
59#4461C2211yescharacteristic
60#3170—31no—
61#3169—31no—
62#2252—31no—
63#2248—31no—
64#2241—31no—
65#2240—31no—
66#2220—31no—
67#2220—31no—
68#2220—31no—
69#2220—31no—
70#2175—31no—
71#2175—31no—
72#2175—31no—
73#2175D311yescharacteristic, complemented
74C4×M32—31no—
75C2×#235—41no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C3×C4×Q8—21no—
79C3×C4×D4—21no—
80#946—21no—
81#944—21no—
82C2×#216—21no—
83#940—21no—
84#938—21no—
85C2×#213—21no—
86C4×#67—41no—
87C2×Dic12C2311yescharacteristic
88C2×D24C2311yescharacteristic
89C2×D3×C8D411yescharacteristic
90C2×D3×C8C2311yescharacteristic
91#897—21no—
92#896—41no—
93#895—21no—
94#889—41no—
95#888—21no—
96#888—21no—
97#885—41no—
98#884—41no—
99C4×D12—21no—
100#872—41no—
101C42×D3D411yescharacteristic
102C4×Dic6—21no—
103C4×Dic6—41no—
104C3×SD32—22no—
105C2×C3×C16C2311yescharacteristic
106C2×C3×C16C2311yescharacteristic
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#103—21no—
110C3×#102—21no—
111C3×C4×C8C2311yescharacteristic
112#835—21no—
113#832—21no—
114#828—22no—
115#827—22no—
116#821—21no—
117#818C2311yescharacteristic
118#818C2311yescharacteristic
119#816—21no—
120C8×Dic3D411yescharacteristic
121C8×Dic3C2311yescharacteristic
122C2×C3⋊C16C2311yescharacteristic
123C2×C3⋊C16C2311yescharacteristic
124#810—21no—
125#809—21no—
126#807—21no—
127#807—21no—
128C4×C3⋊C8D411yescharacteristic
129#800—22no—
130#798—22no—
131#798—22no—
132#533—61no—
133#532—61no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×SD32—31no—
137C2×SD32C2×D311yescharacteristic
138C2×M32—31no—
139C2×M32—31no—
140#500—31no—
141#500—31no—
142#494—31no—
143#493—31no—
144#487—61no—
145#486—61no—
146#467—31no—
147#467—31no—
148#467—61no—
149#466—31no—
150#466—31no—
151#466—61no—
152C4×Q16—31no—
153C4×Q16—61no—
154C4×Q16C2×D311yescharacteristic
155C4×D8—31no—
156C4×D8—61no—
157C4×D8C2×D311yescharacteristic
158C2×C4×C8—31no—
159#366—31no—
160#366—31no—
161#366—31no—
162#366C2×D311yescharacteristic
163#358—31no—
164#358—31no—
165#358—31no—
166#358C2×D311yescharacteristic
167#357—31no—
168#357—31no—
169#357—31no—
170#357C2×D311yescharacteristic
171#346—31no—
172#346C2×D311yescharacteristic
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×D4—21no—
177C22×C3×C4—41no—
178C2×#67—41no—
179C2×#67—41no—
180C2×#67—41no—
181#235—84no—
182C2×D12—21no—
183C2×D12—41no—
184C2×C4×D3—41no—
185C2×C4×D3D811yescharacteristic
186C2×C4×D3D811yescharacteristic
187C2×C4×D3C2×D411yescharacteristic
188C2×Dic6—21no—
189C2×Dic6—41no—
190C2×Dic6—41no—
191C3×Q16—21no—
192C3×Q16—21no—
193C3×D8—21no—
194C3×D8—21no—
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2411yescharacteristic
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—41no—
200C3×C22⋊C4—41no—
201C3×C42—41no—
202C3×C42C2×D411yescharacteristic
203#217—41no—
204#216—42no—
205#213—42no—
206(C2×D3)⋊C4—41no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209(C2×D3)⋊C4—41no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—41no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219C4×Dic3—41no—
220C4×Dic3D811yescharacteristic
221C4×Dic3D811yescharacteristic
222C4×Dic3C2×D411yescharacteristic
223C2×C3⋊C8C2×D411yescharacteristic
224C2×C3⋊C8C2×D411yescharacteristic
225Dic12—21no—
226Dic12—21no—
227D24—21no—
228D24—21no—
229D3×C8—21no—
230D3×C8—21no—
231D3×C8—22no—
232C3×C16—21no—
233C3×C16#4112yescomplemented
234C3⋊C16—21no—
235C3⋊C16#4112yescomplemented
236C2×#41—121no—
237C2×Q16—31no—
238C2×Q16—61no—
239C2×Q16C22×D311yescharacteristic
240C2×D8—31no—
241C2×D8—61no—
242C2×D8C22×D311yescharacteristic
243C22×C8—31no—
244C22×C8—31no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248#127—61no—
249#125—121no—
250#124—121no—
251C22⋊Q8—61no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—61no—
255#121—61no—
256#121—121no—
257C4×Q8—21no—
258C4×Q8—61no—
259C4×Q8—121no—
260C4×D4—21no—
261C4×D4—61no—
262C4×D4—121no—
263#117—121no—
264C2×C42—31no—
265SD32—22no—
266SD32—62no—
267SD32—62no—
268SD32—62no—
269M32—62no—
270M32—62no—
271C2×C16—31no—
272C2×C16—31no—
273C2×C16C22×D311yescharacteristic
274C2×C16C22×D311yescharacteristic
275#107—31no—
276#107—31no—
277#107—61no—
278#107—61no—
279#107C22×D311yescharacteristic
280#107C22×D311yescharacteristic
281#103—21no—
282#103—61no—
283#103—61no—
284#103—61no—
285#102—21no—
286#102—61no—
287#102—61no—
288#102—61no—
289C4×C8—31no—
290C4×C8—31no—
291C4×C8—31no—
292C4×C8C22×D311yescharacteristic
293C23×C3—41no—
294C22×D3—41no—
295C22×D3C2×D811yescharacteristic
296C3×Q8—41no—
297C3×Q8—41no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—82no—
304C2×C3×C4C2×D811yescharacteristic
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307#67—82no—
308#67—82no—
309#67—82no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3C2×D811yescharacteristic
314C2×Dic3C2×D811yescharacteristic
315C2×Dic3C2×D811yescharacteristic
316D12—41no—
317D12—41no—
318D12—82no—
319C4×D3—21no—
320C4×D3—21no—
321C4×D3—22no—
322C4×D3—22no—
323C4×D3—82no—
324Dic6—41no—
325Dic6—41no—
326Dic6—82no—
327Dic6—82no—
328C3×C8—21no—
329C3×C8C2×#4111yescharacteristic
330C3×C8C2×#4111yescharacteristic
331C3⋊C8—21no—
332C3⋊C8#13512yes—
333#41—244no—
334C2×Q8—21no—
335C2×Q8—61no—
336C2×Q8—121no—
337C2×Q8—121no—
338C2×D4—21no—
339C2×D4—61no—
340C2×D4—121no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—41no—
348C22×C4—121no—
349Q16—21no—
350Q16—21no—
351Q16—61no—
352Q16—61no—
353Q16—122no—
354D8—21no—
355D8—21no—
356D8—61no—
357D8—61no—
358D8—122no—
359C2×C8—31no—
360C2×C8—31no—
361C2×C8—62no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8D3×D411yescharacteristic
365C2×C8C23×D311yescharacteristic
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C4⋊C4—121no—
376C4⋊C4—121no—
377C4⋊C4—121no—
378C22⋊C4—41no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C42—31no—
385C42—31no—
386C42—31no—
387C42—41no—
388C42—121no—
389C42D3×D411yescharacteristic
390C16—21no—
391C16—32no—
392C16—61no—
393C16#23512yescomplemented
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3C22×D811yescharacteristic
398C2×D3—21no—
399C2×D3—21no—
400C2×D3—21no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4#53211yescharacteristic
410C3×C4C2×#13511yescharacteristic
411Dic3—21no—
412Dic3—81no—
413Dic3—81no—
414Dic3—81no—
415Dic3#50912yes—
416Dic3#51012yes—
417C23—31no—
418C23—41no—
419C23—121no—
420Q8—41no—
421Q8—41no—
422Q8—121no—
423Q8—121no—
424Q8—242no—
425Q8—242no—
426D4—41no—
427D4—41no—
428D4—121no—
429D4—121no—
430D4—242no—
431D4—242no—
432D4—242no—
433D4—242no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—82no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—242no—
449C2×C4D3×D811yescharacteristic
450C2×C4D3×D811yescharacteristic
451C2×C4C2×D3×D411yescharacteristic
452C8—21no—
453C8—32no—
454C8—61no—
455C8C2×#23511yescharacteristic
456C8D3×#4111yescharacteristic
457C2×C3—161no—
458C2×C3#316911yescharacteristic
459C2×C3C2×#50911yescharacteristic
460C2×C3C2×#51011yescharacteristic
461D3—41no—
462D3—161no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22C2×D3×D811yescharacteristic
473C4—21no—
474C4—21no—
475C4—32no—
476C4—32no—
477C4—61no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—241no—
482C4—241no—
483C4—241no—
484C4#554811yescharacteristic
485C4D3×#13511yescharacteristic
486C3#3375111yescharacteristic, complemented, Sylow
487C2—121no—
488C2—161no—
489C2—481no—
490C2#7959811yescharacteristic
491C2D3×#50911yescharacteristic
492C2D3×#51011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2175C2
socleC22×C3C22×D8
prime radical#79598C2
hypercenter#2175D3
hypocenterC3#33751
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
1. #79725C2semidirect
3. C3×#2175C2semidirect
4. #71488C2semidirect
5. #71479C2semidirect
6. #70913C2semidirect
7. #70884C2semidirect
8. #69329C2semidirect
9. #69320C2semidirect
10. #69230C2semidirect
11. #69215C2semidirect
12. #69198C2semidirect
13. #69183C2semidirect
14. #69077C2semidirect
15. #69071C2semidirect
73. #2175D3semidirect
233. C3×C16#41semidirect
235. C3⋊C16#41semidirect
393. C16#235semidirect
486. C3#33751semidirect

Maximal subgroups

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

subgroupindex
#797252
#795982
C3×#21752
#714882
#714792
#709132
#708842
#693292
#693202
#692302
#692152
#691982
#691832
#690772
#690712
#337513