Finite Group Databasebeta

← #10950897

Subgroups — #10950897

Order 768.

516 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79715C211yesmaximal, characteristic, complemented
2#79583C211yesmaximal, characteristic, complemented
3C3×#2175C211yesmaximal, characteristic, complemented
4#71488C211yesmaximal, characteristic, complemented
5#71480C211yesmaximal, characteristic, complemented
6#70909C211yesmaximal, characteristic, complemented
7#70885C211yesmaximal, characteristic, complemented
8#69329C211yesmaximal, characteristic, complemented
9#69321C211yesmaximal, characteristic, complemented
10#69229C211yesmaximal, characteristic, complemented
11#69216C211yesmaximal, characteristic, complemented
12#69197C211yesmaximal, characteristic
13#69184C211yesmaximal, characteristic, complemented
14#69077C211yesmaximal, characteristic, complemented
15#69071C211yesmaximal, characteristic, complemented
16#33750—31nomaximal, Sylow
17#5580—21no—
18#5557—21no—
19C2×C3×SD32C2211yescharacteristic
20C3×C4×Q16C2211yescharacteristic
21C3×C4×D8C2211yescharacteristic
22#5191C2211yescharacteristic
23#5188C2211yescharacteristic
24C2×#829C2211yescharacteristic
25#5157C2211yescharacteristic
26#5155C2211yescharacteristic
27C2×#826C2211yescharacteristic
28C4×#215—21no—
29C4×#214—21no—
30C2×#800C2211yescharacteristic
31C2×#798C2211yescharacteristic
32C2×#798C2211yescharacteristic
33#4887C2211yescharacteristic
34#4883C2211yescharacteristic
35#4878C2211yescharacteristic
36#4878C2211yescharacteristic
37#4807—21no—
38#4800—21no—
39#4778—21no—
40#4768—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#4490C2211yescharacteristic
58#4490C2211yescharacteristic
59#4461C2211yescharacteristic
60#3168—31no—
61#3168—31no—
62#2253—31no—
63#2249—31no—
64#2242—31no—
65#2242—31no—
66#2220—31no—
67#2220—31no—
68#2219—31no—
69#2218—31no—
70#2177—31no—
71#2176—31no—
72#2175—31no—
73#2175D311yescharacteristic, complemented
74C4×M32—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×D8C2311yescharacteristic
77C3×C4×Q8—21no—
78C3×C4×D4—21no—
79#947—21no—
80#946—21no—
81C2×#215—21no—
82#938—21no—
83#936—21no—
84C2×#214—21no—
85C4×#67—41no—
86#926—41no—
87C2×Dic12C2311yescharacteristic
88C2×D24C2311yescharacteristic
89C2×D3×C8D411yescharacteristic
90C2×D3×C8C2311yescharacteristic
91#898—41no—
92#897—21no—
93#895—21no—
94#894—41no—
95#890—21no—
96#890—21no—
97#883—41no—
98#879—41no—
99C4×D12—21no—
100C4×D12—41no—
101#872—41no—
102C42×D3D411yescharacteristic
103C4×Dic6—21no—
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#829—22no—
115#826—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#808—21no—
127#808—21no—
128C4×C3⋊C8D411yescharacteristic
129#800—22no—
130#798—22no—
131#798—22no—
132#529—61no—
133#529—61no—
134C2×Q32—31no—
135C2×SD32—31no—
136C2×SD32C2×D311yescharacteristic
137C2×D16—31no—
138C2×M32—31no—
139C2×M32—31no—
140#499—31no—
141#499—31no—
142#495—31no—
143#495—31no—
144#488—61no—
145#488—61no—
146#467—31no—
147#467—31no—
148#466—31no—
149#466—31no—
150#465—61no—
151#465—61no—
152C4×Q16—31no—
153C4×Q16C2×D311yescharacteristic
154C4×SD16—61no—
155C4×SD16—61no—
156C4×D8—31no—
157C4×D8C2×D311yescharacteristic
158C2×C4×C8—31no—
159#367—31no—
160#367—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×D12—21no—
180C2×D12—41no—
181C2×C4×D3—41no—
182C2×C4×D3D811yescharacteristic
183C2×C4×D3D811yescharacteristic
184C2×C4×D3C2×D411yescharacteristic
185C2×Dic6—21no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×D8—21no—
189C3×D8—21no—
190C2×C3×C8C2×D411yescharacteristic
191C2×C3×C8C2411yescharacteristic
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—41no—
195C3×C22⋊C4—41no—
196C3×C42—41no—
197C3×C42C2×D411yescharacteristic
198#217—41no—
199#217—41no—
200#217—41no—
201#215—42no—
202#214—42no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213C4⋊Dic3—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218C4×Dic3—41no—
219C4×Dic3D811yescharacteristic
220C4×Dic3D811yescharacteristic
221C4×Dic3C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223C2×C3⋊C8C2×D411yescharacteristic
224Dic12—21no—
225Dic12—21no—
226D24—21no—
227D24—21no—
228D3×C8—21no—
229D3×C8—21no—
230D3×C8—22no—
231C3×C16—21no—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×Q16C22×D311yescharacteristic
237C2×SD16—61no—
238C2×SD16—61no—
239C2×D8—31no—
240C2×D8C22×D311yescharacteristic
241C22×C8—31no—
242C22×C8—31no—
243#126—121no—
244#126—121no—
245#125—61no—
246#125—61no—
247#124—61no—
248#124—61no—
249#123—121no—
250#123—121no—
251C22⋊Q8—61no—
252C22⋊Q8—61no—
253#121—61no—
254#121—61no—
255C4×Q8—21no—
256C4×Q8—61no—
257C4×D4—21no—
258C4×D4—61no—
259C4×D4—121no—
260C4×D4—121no—
261#117—121no—
262#117—121no—
263C2×C42—31no—
264Q32—62no—
265SD32—22no—
266SD32—62no—
267D16—62no—
268M32—62no—
269M32—62no—
270C2×C16—31no—
271C2×C16—31no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107C22×D311yescharacteristic
277#107C22×D311yescharacteristic
278#106—61no—
279#106—61no—
280#103—21no—
281#103—61no—
282#103—61no—
283#103—61no—
284#102—21no—
285#102—61no—
286#102—61no—
287#102—61no—
288C4×C8—31no—
289C4×C8—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C22×D3—41no—
294C22×D3C2×D811yescharacteristic
295C3×Q8—41no—
296C3×Q8—41no—
297C3×D4—41no—
298C3×D4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—82no—
303C2×C3×C4C2×D811yescharacteristic
304C2×C3×C4C2×D811yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306#67—82no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3C2×D811yescharacteristic
311C2×Dic3C2×D811yescharacteristic
312C2×Dic3C2×D811yescharacteristic
313D12—41no—
314D12—41no—
315D12—82no—
316C4×D3—21no—
317C4×D3—21no—
318C4×D3—41no—
319C4×D3—41no—
320C4×D3—82no—
321Dic6—41no—
322Dic6—41no—
323C3×C8—21no—
324C3×C8C2×#4111yescharacteristic
325C3×C8C2×#4111yescharacteristic
326C3⋊C8—21no—
327C3⋊C8#13512yes—
328C2×Q8—21no—
329C2×Q8—61no—
330C2×D4—21no—
331C2×D4—61no—
332C2×D4—121no—
333C2×D4—121no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—41no—
338C22×C4—121no—
339Q16—21no—
340Q16—21no—
341Q16—61no—
342Q16—61no—
343SD16—122no—
344SD16—122no—
345D8—21no—
346D8—21no—
347D8—61no—
348D8—61no—
349C2×C8—31no—
350C2×C8—31no—
351C2×C8—62no—
352C2×C8—61no—
353C2×C8—61no—
354C2×C8D3×D411yescharacteristic
355C2×C8C23×D311yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C22⋊C4—41no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C42—31no—
379C42—31no—
380C42—31no—
381C42—41no—
382C42—121no—
383C42D3×D411yescharacteristic
384C16—21no—
385C16—32no—
386C16—61no—
387C16#23512yes—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3C22×D811yescharacteristic
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—81no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4#53211yescharacteristic
404C3×C4C2×#13511yescharacteristic
405Dic3—21no—
406Dic3—21no—
407Dic3—21no—
408Dic3—81no—
409Dic3—81no—
410Dic3—81no—
411C23—31no—
412C23—41no—
413C23—121no—
414Q8—41no—
415Q8—41no—
416Q8—121no—
417Q8—121no—
418D4—41no—
419D4—41no—
420D4—121no—
421D4—121no—
422D4—242no—
423D4—242no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—82no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—242no—
439C2×C4D3×D811yescharacteristic
440C2×C4D3×D811yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C8—21no—
443C8—32no—
444C8—61no—
445C8C2×#23511yescharacteristic
446C8D3×#4111yescharacteristic
447C2×C3—161no—
448C2×C3#316911yescharacteristic
449C2×C3#341411yescharacteristic
450C2×C3#341411yescharacteristic
451D3—41no—
452D3—161no—
453C22—61no—
454C22—61no—
455C22—61no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22C2×D3×D811yescharacteristic
463C4—21no—
464C4—21no—
465C4—61no—
466C4—61no—
467C4—61no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—241no—
472C4—241no—
473C4—241no—
474C4#554811yescharacteristic
475C4D3×#13511yescharacteristic
476C3#3375011yescharacteristic, complemented, Sylow
477C2—121no—
478C2—161no—
479C2—481no—
480C2#7959811yescharacteristic
481C2#8203511yescharacteristic
482C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2175C2
socleC22×C3C22×D8
prime radical#69197C2
hypercenter#2175D3
hypocenterC3#33750
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. #79715C2semidirect
2. #79583C2semidirect
3. C3×#2175C2semidirect
4. #71488C2semidirect
5. #71480C2semidirect
6. #70909C2semidirect
7. #70885C2semidirect
8. #69329C2semidirect
9. #69321C2semidirect
10. #69229C2semidirect
11. #69216C2semidirect
13. #69184C2semidirect
14. #69077C2semidirect
15. #69071C2semidirect
73. #2175D3semidirect
476. C3#33750semidirect

Maximal subgroups

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

subgroupindex
#797152
#795832
C3×#21752
#714882
#714802
#709092
#708852
#693292
#693212
#692292
#692162
#691972
#691842
#690772
#690712
#337503