Finite Group Databasebeta

← #10844970

Subgroups — #10844970

Order 768.

712 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3139C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#83330C211yesmaximal, characteristic, complemented
4C2×#5083C211yesmaximal, characteristic
5#71789C211yesmaximal, characteristic, complemented
6#70339C212yesmaximal
7#70338C212yesmaximal, complemented
8#70337C212yesmaximal, complemented
9#70335C212yesmaximal
10#69875C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic
12#25313—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C22×#215C2212yes—
16C2×C4×D12C2211yescharacteristic
17C3×#491—22no—
18C3×#487—21no—
19C3×#487—21no—
20C3×#419C2212yes—
21C3×#419C2212yes—
22C3×C4×M16C2211yescharacteristic
23C2×#836C2212yes—
24C2×#836C2212yes—
25C2×#835C2212yes—
26C2×#823C2211yescharacteristic
27C2×#823C2211yescharacteristic
28C2×#822C2212yes—
29#5087—21no—
30#5084—21no—
31#5083—21no—
32#5083—22no—
33#5083—21no—
34#5066—22no—
35#5024—21no—
36#5024—22no—
37Q8×C3⋊C8—21no—
38#5001C2212yes—
39#4999C2212yes—
40#4997C2212yes—
41#4995C2212yes—
42#4924C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C4×#208C2211yescharacteristic
45#4558C2212yes—
46#4554C2211yescharacteristic
47#4554C2211yescharacteristic
48#3139D311yescharacteristic, complemented
49#3077—31no—
50C2×#461—31no—
51#2962—31no—
52#2064—32no—
53#2046—32no—
54#1977—32no—
55#1966—31no—
56#1950—31no—
57#1930—32no—
58C4×#100—31no—
59C22×C3×Q8C2312yes—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C2×C3×M16C2311yescharacteristic
63C2×C3×M16C2311yescharacteristic
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8D411yescharacteristic
69C3×C4⋊Q8D411yescharacteristic
70C3×#125D412yes—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×#117C2312yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C42C2311yescharacteristic
79C2×#215—24no—
80C2×#215—22no—
81C2×#215—22no—
82C2×#215—42no—
83C2×(C2×D3)⋊C4—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×#208C2312yes—
86C2×#208C2311yescharacteristic
87C2×#208C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89C4×D12—21no—
90C4×D12—21no—
91C4×D12—21no—
92C4×D12—21no—
93C3×#103—22no—
94C3×#103—22no—
95C3×#103—22no—
96C3×#103—22no—
97C3×#97—21no—
98C3×C4×C8—21no—
99#836—24no—
100#836—42no—
101#835—22no—
102#835—22no—
103#823—22no—
104#823—22no—
105#822—42no—
106#809—22no—
107#809—22no—
108#808—22no—
109#807—22no—
110C4⋊(C3⋊C8)—21no—
111C4⋊(C3⋊C8)—21no—
112C4⋊(C3⋊C8)—22no—
113C4⋊(C3⋊C8)—22no—
114C4⋊(C3⋊C8)D411yescharacteristic
115C4⋊(C3⋊C8)D412yes—
116C4⋊(C3⋊C8)D411yescharacteristic
117#803—21no—
118#803—22no—
119C4×C3⋊C8—21no—
120C4×C3⋊C8—22no—
121C22×SD16—32no—
122#533C2×D311yescharacteristic
123C2×C4⋊Q8C2×D311yescharacteristic
124C2×C4×D4—31no—
125#491—22no—
126#487—21no—
127#487—21no—
128#478—62no—
129#476—61no—
130#474—61no—
131#461—61no—
132#461—62no—
133#461—61no—
134#446—62no—
135#446—61no—
136C8×Q8—61no—
137C2×#108—32no—
138#428—32no—
139#424—32no—
140#423—31no—
141C2×C4⋊C8—31no—
142#419C2×D312yes—
143#419C2×D312yes—
144#418—32no—
145C2×#103—32no—
146C2×#101—32no—
147C2×#101—32no—
148C2×#100—31no—
149C2×#100—31no—
150#405—32no—
151C4×M16—31no—
152C4×M16C2×D311yescharacteristic
153#343—31no—
154#343—31no—
155#341—32no—
156C23×D3—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C22×C3×C4—22no—
166C22×C3×C4#4111yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12#4112yes—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×C4×D3—82no—
179C3×M16—21no—
180C3×M16—21no—
181C3×M16—21no—
182C3×M16—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4—41no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4C2×D412yes—
194C3×C4⋊C4C2×D412yes—
195C3×C22⋊C4—22no—
196C3×C42—22no—
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201#215—44no—
202#215—44no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207#208—21no—
208#208—21no—
209#208—22no—
210#208—22no—
211#208—21no—
212#208#4112yes—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8C2×D412yes—
219C2×C3⋊C8C2×D412yes—
220C22×Q8C22×D312yes—
221C22×D4—31no—
222C23×C4—61no—
223C2×SD16—62no—
224C2×SD16—62no—
225C2×SD16—64no—
226C2×SD16—122no—
227C2×M16—31no—
228C2×M16—32no—
229C2×M16—31no—
230C2×M16C22×D311yescharacteristic
231C2×M16C22×D311yescharacteristic
232C22×C8—32no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8#6711yescharacteristic
238C4⋊Q8#6711yescharacteristic
239#125#6712yes—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×D4—61no—
245C4×D4—61no—
246C4×D4—61no—
247C4×D4—61no—
248#117C22×D312yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C22⋊C4—61no—
253C2×C42C22×D311yescharacteristic
254#108—122no—
255#107—62no—
256#106—62no—
257C4⋊C8—31no—
258C4⋊C8—31no—
259C4⋊C8—32no—
260C4⋊C8—62no—
261C4⋊C8—62no—
262C4⋊C8—61no—
263C4⋊C8—61no—
264#103—22no—
265#103—22no—
266#103—22no—
267#103—22no—
268#103—62no—
269#103—62no—
270#102—62no—
271#102—62no—
272#101—64no—
273#101—122no—
274#100—62no—
275#100—62no—
276#97—21no—
277#97—62no—
278#97—61no—
279C4×C8—21no—
280C4×C8—62no—
281C4×C8—61no—
282C23×C3C2×#4111yescharacteristic
283C22×D3—22no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—81no—
287C22×D3—81no—
288C3×Q8—42no—
289C3×Q8—42no—
290C3×Q8—42no—
291C3×Q8—42no—
292C3×Q8—42no—
293C3×Q8—81no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×Dic3—41no—
312C2×Dic3—81no—
313D12—22no—
314D12—22no—
315D12—41no—
316D12—41no—
317C4×D3—82no—
318C4×D3—82no—
319C3×C8—41no—
320C3×C8—41no—
321C3⋊C8—22no—
322C3⋊C8—22no—
323C3⋊C8—22no—
324C3⋊C8—41no—
325C3⋊C8—42no—
326C24—61no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×D4—32no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4—61no—
340C2×D4—61no—
341C22×C4—22no—
342C22×C4—61no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4—242no—
346C22×C4#23711yescharacteristic
347C22×C4#23711yescharacteristic
348C22×C4C23×D311yescharacteristic
349SD16—124no—
350SD16—124no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—32no—
356M16—61no—
357M16—61no—
358M16—62no—
359M16—61no—
360M16—62no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—32no—
364C2×C8—32no—
365C2×C8—61no—
366C2×C8—62no—
367C2×C8—62no—
368C2×C8—61no—
369C2×C8—62no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4C2×#6712yes—
381C4⋊C4C2×#6712yes—
382C22⋊C4—22no—
383C22⋊C4—121no—
384C22⋊C4—121no—
385C42—22no—
386C42D3×D411yescharacteristic
387C42D3×D411yescharacteristic
388C42C2×#6711yescharacteristic
389C42C2×#6711yescharacteristic
390C22×C3—21no—
391C22×C3—21no—
392C22×C3C2×#12111yescharacteristic
393C22×C3#52911yescharacteristic
394C22×C3#54211yescharacteristic
395C2×D3—42no—
396C2×D3—82no—
397C2×D3—81no—
398C2×D3—81no—
399C2×D3—82no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—42no—
404C3×C4—42no—
405C3×C4—82no—
406C3×C4#47112yes—
407C3×C4#54611yescharacteristic
408C3×C4#54811yescharacteristic
409C3×C4#54811yescharacteristic
410C3×C4#55011yescharacteristic
411Dic3—81no—
412C23—62no—
413C23—121no—
414C23—121no—
415C23—241no—
416C23—241no—
417C23C2×#23711yescharacteristic
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—81no—
424D4—62no—
425D4—62no—
426D4—121no—
427D4—121no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—121no—
436C2×C4—242no—
437C2×C4—242no—
438C2×C4—241no—
439C2×C4#93911yescharacteristic
440C2×C4#93911yescharacteristic
441C2×C4#93911yescharacteristic
442C2×C4#93911yescharacteristic
443C2×C4C2×D3×D411yescharacteristic
444C2×C4D3×#4111yescharacteristic
445C2×C4D3×#4111yescharacteristic
446C2×C4#100911yescharacteristic
447C2×C4#101011yescharacteristic
448C2×C4C22×#6711yescharacteristic
449C8—41no—
450C8—41no—
451C8—62no—
452C8—62no—
453C8—62no—
454C8—121no—
455C8—122no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3#242611yescharacteristic
459C2×C3C2×#47111yescharacteristic
460C2×C3#306911yescharacteristic
461D3—82no—
462C22—21no—
463C22—21no—
464C22—122no—
465C22—242no—
466C22—242no—
467C22—241no—
468C22—241no—
469C22#568011yescharacteristic
470C22#568211yescharacteristic
471C22C2×#93911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—241no—
479C4#520312yes—
480C4#567611yescharacteristic
481C4#567711yescharacteristic
482C4#583111yescharacteristic
483C4#583411yescharacteristic
484C3#2531311yescharacteristic, complemented, Sylow
485C2—21no—
486C2—21no—
487C2—242no—
488C2#8299511yescharacteristic
489C2C2×#520311yescharacteristic
490C2#8397811yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3139C2
socleC22×C3C2×#121
prime radicalC22×D12C23
hypercenter#3139D3
hypocenterC3#25313
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
1. C3×#3139C2semidirect
3. #83330C2semidirect
5. #71789C2semidirect
7. #70338C2semidirect
8. #70337C2semidirect
10. #69875C2semidirect
48. #3139D3semidirect
484. C3#25313semidirect

Maximal subgroups

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

subgroupindex
C3×#31392
#833502
#833302
C2×#50832
#717892
#703392
#703382
#703372
#703352
#698752
C4×#8232
#253133