Finite Group Databasebeta

← #10950903

Subgroups — #10950903

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#80741C211yesmaximal, characteristic, complemented
2#80697C211yesmaximal, characteristic
3C3×#2220C211yesmaximal, characteristic, complemented
4#71479C211yesmaximal, characteristic, complemented
5#71474C211yesmaximal, characteristic, complemented
6#71444C211yesmaximal, characteristic, complemented
7Dic3×M32C211yesmaximal, characteristic, complemented
8#71084C211yesmaximal, characteristic, complemented
9#71074C211yesmaximal, characteristic, complemented
10#69320C211yesmaximal, characteristic, complemented
11#69319C211yesmaximal, characteristic, complemented
12#69215C211yesmaximal, characteristic, complemented
13#69208C211yesmaximal, characteristic, complemented
14#69183C211yesmaximal, characteristic, complemented
15#69176C211yesmaximal, characteristic, complemented
16#33751—31nomaximal, Sylow
17#5622—21no—
18#5596—21no—
19C2×C3×SD32C2211yescharacteristic
20C2×C3×M32C2211yescharacteristic
21C3×#467C2211yescharacteristic
22C3×#466C2211yescharacteristic
23#5183C2211yescharacteristic
24Dic3×Q16C2211yescharacteristic
25C2×#828C2211yescharacteristic
26#5151C2211yescharacteristic
27Dic3×D8C2211yescharacteristic
28C2×#827C2211yescharacteristic
29#5116C2211yescharacteristic
30#5115C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32C2×#812C2211yescharacteristic
33#5051—21no—
34#5038—21no—
35C2×#800C2211yescharacteristic
36#4874C2211yescharacteristic
37#4874C2211yescharacteristic
38#4873C2211yescharacteristic
39#4872C2211yescharacteristic
40#4792—21no—
41#4790—21no—
42#4760—21no—
43#4756—21no—
44C3×#366C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#357C2211yescharacteristic
47#4564C2211yescharacteristic
48#4562C2211yescharacteristic
49#4509C2211yescharacteristic
50#4504C2211yescharacteristic
51#4503C2211yescharacteristic
52#4502C2211yescharacteristic
53#4502C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55Dic3×C16C2211yescharacteristic
56#4492C2211yescharacteristic
57#4491C2211yescharacteristic
58#4489C2211yescharacteristic
59#4489C2211yescharacteristic
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#2220D311yescharacteristic, complemented
70#2175—31no—
71#2175—31no—
72#2175—31no—
73#2175—31no—
74C4×M32—31no—
75C2×#237—41no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C22⋊Q8—21no—
80C3×#121—21no—
81Q8×Dic3—21no—
82#944—21no—
83C2×#216—21no—
84#940—21no—
85#939—41no—
86D4×Dic3—21no—
87C2×#213—21no—
88#930—21no—
89#924—21no—
90C2×C4×Dic3D411yescharacteristic
91C22×C3⋊C8D411yescharacteristic
92C2×Dic12C2311yescharacteristic
93C2×D24C2311yescharacteristic
94#893—21no—
95#889—41no—
96#888—21no—
97#888—21no—
98#887—21no—
99#887—41no—
100#885—41no—
101#67⋊C4—41no—
102#878—41no—
103#877—41no—
104C3×SD32—22no—
105C3×M32—22no—
106C2×C3×C16C2311yescharacteristic
107C2×C3×C16C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#107C2311yescharacteristic
110C3×#103—21no—
111C3×#102—21no—
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×Dic3D411yescharacteristic
122C8×Dic3C2311yescharacteristic
123C8×Dic3C2311yescharacteristic
124#812—22no—
125C2×C3⋊C16C2311yescharacteristic
126C2×C3⋊C16C2311yescharacteristic
127#810—21no—
128#809—21no—
129#807—21no—
130#807—21no—
131#800—22no—
132#533—61no—
133#532—61no—
134C2×SD32—31no—
135C2×SD32—31no—
136C2×SD32—31no—
137C2×SD32C2×D311yescharacteristic
138C2×M32—31no—
139C2×M32C2×D311yescharacteristic
140#500—31no—
141#500—31no—
142#494—31no—
143#493—31no—
144#487—61no—
145#486—61no—
146#467—31no—
147#467—61no—
148#467C2×D311yescharacteristic
149#466—31no—
150#466—61no—
151#466C2×D311yescharacteristic
152C4×Q16—31no—
153C4×Q16—31no—
154C4×Q16—61no—
155C4×D8—31no—
156C4×D8—31no—
157C4×D8—61no—
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#346—31no—
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C22×C3×C4C2×D411yescharacteristic
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C22×Dic3—41no—
183C22×Dic3D811yescharacteristic
184C22×Dic3D811yescharacteristic
185#237—84no—
186C2×D12—21no—
187C2×C4×D3—41no—
188C2×Dic6—21no—
189C2×Dic6—41no—
190C2×Dic6—41no—
191C3×Q16—21no—
192C3×Q16—21no—
193C3×D8—21no—
194C3×D8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2411yescharacteristic
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—41no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—41no—
204#217—41no—
205#217—41no—
206#216—42no—
207#213—42no—
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×Dic3—41no—
221C4×Dic3D811yescharacteristic
222C4×Dic3D811yescharacteristic
223C4×Dic3C2×D411yescharacteristic
224C4×Dic3C2×D411yescharacteristic
225C2×C3⋊C8—22no—
226C2×C3⋊C8C2×D411yescharacteristic
227C2×C3⋊C8C2×D411yescharacteristic
228Dic12—21no—
229Dic12—21no—
230D24—21no—
231D24—21no—
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×C8C22×D311yescharacteristic
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248#127—61no—
249#125—121no—
250#124—121no—
251C22⋊Q8—21no—
252C22⋊Q8—61no—
253C22⋊Q8—121no—
254#121—21no—
255#121—61no—
256#121—121no—
257C4×Q8—61no—
258C4×Q8—61no—
259C4×Q8—121no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—121no—
263#117—121no—
264C2×C42—31no—
265SD32—22no—
266SD32—62no—
267SD32—62no—
268SD32—62no—
269M32—22no—
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×C8—31no—
293C23×C3—41no—
294C23×C3C2×D811yescharacteristic
295C22×D3—41no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—82no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308#67—82no—
309#67—82no—
310#67—82no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—82no—
317C2×Dic3C2×D811yescharacteristic
318C2×Dic3C2×D811yescharacteristic
319C2×Dic3C2×D811yescharacteristic
320C2×Dic3C2×D811yescharacteristic
321D12—41no—
322D12—41no—
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—41no—
340C2×D4—61no—
341C2×D4—121no—
342C2×D4—121no—
343C2×D4—121no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—121no—
347C22×C4—121no—
348C22×C4D3×D411yescharacteristic
349Q16—21no—
350Q16—21no—
351Q16—61no—
352Q16—61no—
353Q16—122no—
354D8—21no—
355D8—21no—
356D8—61no—
357D8—61no—
358D8—122no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—31no—
362C2×C8—31no—
363C2×C8—62no—
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—41no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C42—31no—
385C42—31no—
386C42—31no—
387C42—31no—
388C42—121no—
389C42—121no—
390C16—21no—
391C16—32no—
392C16—61no—
393C16#23712yescomplemented
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3C22×D811yescharacteristic
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—21no—
405C3×C4—81no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4#53211yescharacteristic
409C3×C4C2×#13511yescharacteristic
410Dic3—21no—
411Dic3—21no—
412Dic3—81no—
413Dic3—81no—
414Dic3—81no—
415Dic3#50912yes—
416Dic3#51012yes—
417C23—41no—
418C23—121no—
419C23D3×D811yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—121no—
423Q8—121no—
424Q8—242no—
425Q8—242no—
426D4—41no—
427D4—41no—
428D4—82no—
429D4—121no—
430D4—121no—
431D4—242no—
432D4—242no—
433D4—242no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—242no—
449C2×C4—242no—
450C2×C4D3×D811yescharacteristic
451C2×C4C2×D3×D411yescharacteristic
452C8—21no—
453C8—32no—
454C8—61no—
455C8C2×#23711yescharacteristic
456C8D3×#4111yescharacteristic
457C2×C3—41no—
458C2×C3—161no—
459C2×C3#316911yescharacteristic
460C2×C3C2×#50911yescharacteristic
461C2×C3C2×#51011yescharacteristic
462D3—161no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22C2×D3×D811yescharacteristic
473C4—21no—
474C4—32no—
475C4—32no—
476C4—61no—
477C4—61no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—241no—
482C4—241no—
483C4—241no—
484C4#559611yescharacteristic
485C4D3×#13511yescharacteristic
486C3#3375111yescharacteristic, complemented, Sylow
487C2—41no—
488C2—161no—
489C2—481no—
490C2#8069711yescharacteristic
491C2D3×#50911yescharacteristic
492C2D3×#51011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2220C2
socleC22×C3C22×D8
prime radical#80697C2
hypercenter#2220D3
hypocenterC3#33751
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2220

Complements

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

NH ≅ G/Ntype
1. #80741C2semidirect
3. C3×#2220C2semidirect
4. #71479C2semidirect
5. #71474C2semidirect
6. #71444C2semidirect
7. Dic3×M32C2semidirect
8. #71084C2semidirect
9. #71074C2semidirect
10. #69320C2semidirect
11. #69319C2semidirect
12. #69215C2semidirect
13. #69208C2semidirect
14. #69183C2semidirect
15. #69176C2semidirect
69. #2220D3semidirect
233. C3×C16#41semidirect
235. C3⋊C16#41semidirect
393. C16#237semidirect
486. C3#33751semidirect

Maximal subgroups

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

subgroupindex
#807412
#806972
C3×#22202
#714792
#714742
#714442
Dic3×M322
#710842
#710742
#693202
#693192
#692152
#692082
#691832
#691762
#337513