Finite Group Databasebeta

← #10836668

Subgroups — #10836668

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5021C211yesmaximal, characteristic, complemented
2C4×#942C211yesmaximal, characteristic
3#79060C211yesmaximal, characteristic
4#79040C211yesmaximal, characteristic
5C3×#1801C211yesmaximal, characteristic, complemented
6#70524C211yesmaximal, characteristic, complemented
7#70520C211yesmaximal, characteristic, complemented
8#70126C212yesmaximal
9#70106C212yesmaximal, complemented
10#69010C212yesmaximal
11#68985C212yesmaximal, complemented
12#24722—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#942C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×#869C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C3×#423C2211yescharacteristic
19C2×C3×#101C2212yes—
20#5225C2211yescharacteristic
21C2×#835C2212yes—
22#5133C2212yes—
23#5129C2212yes—
24C2×#822C2212yes—
25#5122C2212yes—
26#5121C2212yes—
27#5030—21no—
28C4×#216—21no—
29#5026—21no—
30C4×#215—21no—
31#5021—21no—
32#5021—21no—
33#5021—22no—
34#4966C2211yescharacteristic
35#4964C2211yescharacteristic
36C2×#807C2212yes—
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C4×#208C2211yescharacteristic
39#4851—21no—
40#4850—21no—
41#4846—21no—
42#4842—21no—
43#4840—22no—
44#4835—21no—
45#4833—21no—
46C3×#343C2212yes—
47#4558C2212yes—
48#4554C2212yes—
49#4554C2212yes—
50#3079—31no—
51C2×#477—31no—
52C4×#137—31no—
53#2925—31no—
54#2078—32no—
55#2052—32no—
56#1997—31no—
57#1952—32no—
58#1950—31no—
59#1930—32no—
60#1801D311yescharacteristic, complemented
61C22×C3×Q8C2311yescharacteristic
62C2×#235C2311yescharacteristic
63C22×Dic6—21no—
64C2×C3×M16C2312yes—
65C2×C3×M16C2312yes—
66C3×C4×Q8—21no—
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#216—21no—
75#942—22no—
76#942—22no—
77C2×#215—21no—
78C4×#67—21no—
79C4×#67—21no—
80#926C2311yescharacteristic
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3C2312yes—
83C2×#208C2311yescharacteristic
84C2×#208C2311yescharacteristic
85C22×C3⋊C8C2312yes—
86C4×D12D411yescharacteristic
87C4×D12D411yescharacteristic
88#872—21no—
89C42×D3—21no—
90#869—21no—
91#869—21no—
92#869—21no—
93#869—21no—
94C4×Dic6D411yescharacteristic
95C4×Dic6D411yescharacteristic
96C3×C4⋊C8—21no—
97C3×C4⋊C8—21no—
98C3×C4⋊C8—21no—
99C3×C4⋊C8—21no—
100C3×#101—24no—
101#835—21no—
102#835—21no—
103#835—22no—
104#835—22no—
105#822—24no—
106#821—22no—
107(C2×D3)⋊C8—22no—
108(C2×D3)⋊C8—22no—
109#818—22no—
110#817—22no—
111#816—22no—
112#815—22no—
113C8×Dic3—22no—
114#810—21no—
115#809—21no—
116#808—21no—
117#807—21no—
118#807—22no—
119#807—22no—
120C4⋊(C3⋊C8)D411yescharacteristic
121C4⋊(C3⋊C8)D411yescharacteristic
122C4⋊(C3⋊C8)D412yes—
123#803—21no—
124C4×C3⋊C8—21no—
125C2×#137—31no—
126C2×C4⋊Q8—31no—
127C4×#41—31no—
128C2×C4×Q8C2×D311yescharacteristic
129#477—61no—
130#477—61no—
131#477—61no—
132#477—62no—
133#476—61no—
134#475—61no—
135#474—61no—
136#441—61no—
137#440—61no—
138C4×Q16—61no—
139C4×SD16—61no—
140#433—62no—
141#432—61no—
142#431—61no—
143C2×#108—32no—
144#428—32no—
145#428—31no—
146C2×#107—32no—
147#423C2×D311yescharacteristic
148#423C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150#419—31no—
151#417—32no—
152#417—31no—
153C2×#103—32no—
154C2×#101C2×D312yes—
155#408—32no—
156#405—32no—
157C4×M16—31no—
158#343—32no—
159#343—32no—
160#343C2×D312yes—
161#341—32no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8#4112yes—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4#4111yescharacteristic
171C22×C3×C4#4111yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C2×#67—21no—
174C22×Dic3—22no—
175#235—21no—
176#235—21no—
177#235—21no—
178#235—21no—
179C2×D12C2×D411yescharacteristic
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×Dic6—41no—
184C2×Dic6—41no—
185C2×Dic6—82no—
186C2×Dic6C2×D411yescharacteristic
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C42—41no—
198C3×C42—41no—
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203#217—21no—
204#216—24no—
205#215—24no—
206(C2×D3)⋊C4—21no—
207(C2×D3)⋊C4—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—41no—
211C4⋊Dic3—41no—
212C4⋊Dic3C2×D411yescharacteristic
213C4⋊Dic3C2×D411yescharacteristic
214Dic3⋊C4—21no—
215Dic3⋊C4—21no—
216C4×Dic3—21no—
217C4×Dic3—21no—
218C4×Dic3—21no—
219#208—21no—
220#208—21no—
221#208—21no—
222#208#4112yes—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8C2×D412yes—
228C2×C3⋊C8C2×D412yes—
229C2×#41—31no—
230C22×Q8—61no—
231C22×Q8C22×D311yescharacteristic
232#137—62no—
233#137—62no—
234C2×Q16—61no—
235C2×SD16—61no—
236C2×M16—31no—
237C2×M16—31no—
238C2×M16C22×D312yes—
239C2×M16C22×D312yes—
240C22×C8—32no—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—31no—
250C4×Q8—31no—
251C4×D4—31no—
252C4×D4—31no—
253C4×D4—61no—
254C4×D4—61no—
255#117—31no—
256#117—61no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C42—21no—
262C2×C42—61no—
263C2×C42C22×D311yescharacteristic
264#108—64no—
265#107—61no—
266#107—62no—
267#107—62no—
268#107—62no—
269#106—61no—
270#106—62no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—31no—
276C4⋊C8—31no—
277C4⋊C8—32no—
278#103—61no—
279#103—61no—
280#103—61no—
281#103—62no—
282#103—62no—
283#103—62no—
284#102—61no—
285#102—62no—
286#101—24no—
287C22⋊C8—62no—
288C22⋊C8—62no—
289#97—62no—
290#97—61no—
291C4×C8—62no—
292C4×C8—61no—
293C23×C3C2×#4111yescharacteristic
294C22×D3—21no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—81no—
306C2×C3×C4#12111yescharacteristic
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12111yescharacteristic
309C2×C3×C4#12111yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316#67—41no—
317#67—41no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—42no—
322C2×Dic3—82no—
323D12—21no—
324D12—21no—
325C4×D3—41no—
326C4×D3—41no—
327C4×D3—42no—
328C4×D3—81no—
329Dic6—21no—
330Dic6—21no—
331Dic6—82no—
332Dic6—82no—
333C3×C8—42no—
334C3×C8—42no—
335C3⋊C8—22no—
336C3⋊C8—22no—
337C3⋊C8—22no—
338C3⋊C8—41no—
339#41—61no—
340#41—61no—
341#41—61no—
342#41—61no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—31no—
349C2×Q8—121no—
350C2×Q8—121no—
351C2×Q8—242no—
352C2×Q8#23712yes—
353C2×D4—31no—
354C2×D4—61no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4#23511yescharacteristic
362C22×C4#23511yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—64no—
365SD16—64no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—32no—
371M16—61no—
372M16—61no—
373M16—61no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—32no—
377C2×C8—32no—
378C2×C8—61no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—61no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—31no—
385C4⋊C4—31no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—121no—
393C4⋊C4—121no—
394C22⋊C4—61no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C42—41no—
398C42—41no—
399C42—61no—
400C42—61no—
401C42—61no—
402C42D3×D411yescharacteristic
403C42D3×D411yescharacteristic
404C42D3×D411yescharacteristic
405C42D3×D411yescharacteristic
406C22×C3—21no—
407C22×C3—21no—
408C22×C3C2×#12111yescharacteristic
409C22×C3#55211yescharacteristic
410C22×C3#55611yescharacteristic
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—41no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—42no—
418C3×C4—81no—
419C3×C4#47312yes—
420C3×C4#52511yescharacteristic
421C3×C4#53611yescharacteristic
422C3×C4#54711yescharacteristic
423C3×C4#54811yescharacteristic
424Dic3—41no—
425Dic3—41no—
426Dic3—41no—
427Dic3—82no—
428C23—61no—
429C23C2×#23511yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—41no—
433Q8—41no—
434Q8—61no—
435Q8—61no—
436Q8—242no—
437Q8—242no—
438D4—61no—
439D4—61no—
440D4—121no—
441D4—121no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—81no—
452C2×C4—122no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—241no—
457C2×C4—242no—
458C2×C4#88411yescharacteristic
459C2×C4#88411yescharacteristic
460C2×C4#88411yescharacteristic
461C2×C4#88411yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×D3×D411yescharacteristic
464C2×C4C2×#23711yescharacteristic
465C2×C4#100711yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C2×C4#101011yescharacteristic
468C8—42no—
469C8—42no—
470C8—62no—
471C8—62no—
472C8—62no—
473C8—121no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3#241511yescharacteristic
477C2×C3C2×#47311yescharacteristic
478C2×C3#306811yescharacteristic
479D3—81no—
480C22—21no—
481C22—21no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22C2×#88411yescharacteristic
486C22#551311yescharacteristic
487C22#553311yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—81no—
493C4—121no—
494C4—121no—
495C4—121no—
496C4—242no—
497C4#475512yes—
498C4#559811yescharacteristic
499C4#560611yescharacteristic
500C4#562811yescharacteristic
501C4#563111yescharacteristic
502C3#2472211yescharacteristic, complemented, Sylow
503C2—21no—
504C2—21no—
505C2—241no—
506C2#7402011yescharacteristic
507C2C2×#475511yescharacteristic
508C2#7872011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1801C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#1801D3
hypocenterC3#24722
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. C2×#5021C2semidirect
5. C3×#1801C2semidirect
6. #70524C2semidirect
7. #70520C2semidirect
9. #70106C2semidirect
11. #68985C2semidirect
60. #1801D3semidirect
502. C3#24722semidirect

Maximal subgroups

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

subgroupindex
C2×#50212
C4×#9422
#790602
#790402
C3×#18012
#705242
#705202
#701262
#701062
#690102
#689852
#247223