Finite Group Databasebeta

← #10836657

Subgroups — #10836657

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5008C211yesmaximal, characteristic, complemented
2C4×#932C211yesmaximal, characteristic
3#79060C211yesmaximal, characteristic, complemented
4#79040C211yesmaximal, characteristic, complemented
5C3×#1800C211yesmaximal, characteristic, complemented
6#70524C211yesmaximal, characteristic, complemented
7#70520C211yesmaximal, characteristic, complemented
8#70092C212yesmaximal
9#70072C212yesmaximal, complemented
10#69011C212yesmaximal, complemented
11#68984C212yesmaximal, complemented
12#24721—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#932C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×#869C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C3×#423C2211yescharacteristic
19C2×C3×#100C2212yes—
20#5215C2211yescharacteristic
21C2×#832C2212yes—
22#5133C2212yes—
23#5129C2212yes—
24C2×#822C2212yes—
25#5122C2212yes—
26#5121C2212yes—
27#5018—21no—
28C4×#214—21no—
29#5014—21no—
30C4×#213—21no—
31#5008—21no—
32#5008—21no—
33#5008—22no—
34#4966C2211yescharacteristic
35#4964C2211yescharacteristic
36C2×#808C2212yes—
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×#476—31no—
52C4×AGL(1,ℤ/8)—31no—
53#2925—31no—
54#2079—32no—
55#2051—32no—
56#1997—31no—
57#1951—32no—
58#1950—31no—
59#1931—32no—
60#1800D311yescharacteristic, complemented
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C2×#235C2311yescharacteristic
64C22×Dic6—21no—
65C2×C3×M16C2312yes—
66C2×C3×M16C2312yes—
67C3×C4×D4—21no—
68C3×C4×D4—21no—
69C3×C4×D4—21no—
70C3×C4×D4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#214—21no—
75#932—22no—
76#932—22no—
77C2×#213—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×#100—24no—
101#832—21no—
102#832—21no—
103#832—22no—
104#832—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#808—22no—
118#808—22no—
119#807—21no—
120C4⋊(C3⋊C8)D411yescharacteristic
121C4⋊(C3⋊C8)D411yescharacteristic
122C4⋊(C3⋊C8)D412yes—
123#803—21no—
124C4×C3⋊C8—21no—
125C2×AGL(1,ℤ/8)—31no—
126C2×C4⋊Q8—31no—
127C4×#41—31no—
128C2×C4×D4C2×D311yescharacteristic
129#477—61no—
130#476—61no—
131#476—61no—
132#476—61no—
133#476—62no—
134#475—61no—
135#474—61no—
136#442—61no—
137#440—61no—
138C4×SD16—61no—
139C4×D8—61no—
140#433—62no—
141#432—61no—
142#431—61no—
143C2×#108—32no—
144#428—32no—
145#428—31no—
146C2×#106—32no—
147#423C2×D311yescharacteristic
148#423C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150#418—31no—
151#417—32no—
152#417—31no—
153C2×#102—32no—
154C2×#100C2×D312yes—
155#408—32no—
156#405—32no—
157C4×M16—31no—
158#343—32no—
159#343—32no—
160#343C2×D312yes—
161#341—32no—
162C24×C3—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4#4112yes—
169C22×C3×C4—21no—
170C22×C3×C4—41no—
171C22×C3×C4—41no—
172C22×C3×C4—82no—
173C22×C3×C4#4111yescharacteristic
174C22×C3×C4#4111yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—21no—
177C22×Dic3—22no—
178#235—21no—
179#235—21no—
180#235—21no—
181#235—21no—
182C2×D12C2×D411yescharacteristic
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×Dic6—41no—
187C2×Dic6—41no—
188C2×Dic6—82no—
189C2×Dic6C2×D411yescharacteristic
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C22⋊C4—41no—
199C3×C22⋊C4—41no—
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#217—21no—
205#214—24no—
206#213—24no—
207(C2×D3)⋊C4—21no—
208(C2×D3)⋊C4—21no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213C4⋊Dic3C2×D411yescharacteristic
214C4⋊Dic3C2×D411yescharacteristic
215Dic3⋊C4—21no—
216Dic3⋊C4—21no—
217C4×Dic3—21no—
218C4×Dic3—21no—
219C4×Dic3—21no—
220#208—21no—
221#208—21no—
222#208—21no—
223#208#4112yes—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8C2×D412yes—
229C2×C3⋊C8C2×D412yes—
230C2×#41—31no—
231C22×Q8—61no—
232C22×D4C22×D311yescharacteristic
233C23×C4—21no—
234AGL(1,ℤ/8)—62no—
235AGL(1,ℤ/8)—62no—
236C2×SD16—61no—
237C2×D8—61no—
238C2×M16—31no—
239C2×M16—31no—
240C2×M16C22×D312yes—
241C2×M16C22×D312yes—
242C22×C8—32no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4×Q8—31no—
248C4×Q8—31no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—31no—
254C4×D4—31no—
255C4×D4—61no—
256C4×D4—61no—
257#117—31no—
258#117—61no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C42—61no—
264C2×C42C22×D311yescharacteristic
265#108—64no—
266#107—61no—
267#107—62no—
268#106—61no—
269#106—62no—
270#106—62no—
271#106—62no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—31no—
277C4⋊C8—31no—
278C4⋊C8—32no—
279#103—61no—
280#103—62no—
281#102—61no—
282#102—61no—
283#102—61no—
284#102—62no—
285#102—62no—
286#102—62no—
287#100—24no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#97—62no—
291#97—61no—
292C4×C8—62no—
293C4×C8—61no—
294C23×C3—22no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—81no—
298C23×C3—81no—
299C23×C3C2×#4111yescharacteristic
300C22×D3—21no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—41no—
304C3×D4—41no—
305C2×C3×C4—22no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—82no—
309C2×C3×C4—82no—
310C2×C3×C4—81no—
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320C2×C3×C42−1+411yescharacteristic
321#67—41no—
322#67—41no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—42no—
327C2×Dic3—82no—
328D12—21no—
329D12—21no—
330C4×D3—41no—
331C4×D3—41no—
332C4×D3—42no—
333C4×D3—81no—
334Dic6—21no—
335Dic6—21no—
336Dic6—82no—
337Dic6—82no—
338C3×C8—42no—
339C3×C8—42no—
340C3⋊C8—22no—
341C3⋊C8—22no—
342C3⋊C8—22no—
343C3⋊C8—41no—
344C24—21no—
345#41—61no—
346#41—61no—
347#41—61no—
348#41—61no—
349C2×Q8—31no—
350C2×Q8—121no—
351C2×Q8—121no—
352C2×Q8—242no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—31no—
359C2×D4—61no—
360C2×D4#23712yes—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—82no—
369C22×C4#23511yescharacteristic
370C22×C4#23511yescharacteristic
371C22×C4C23×D311yescharacteristic
372SD16—64no—
373D8—64no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—22no—
378M16—32no—
379M16—61no—
380M16—61no—
381M16—61no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—32no—
385C2×C8—32no—
386C2×C8—61no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—61no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—31no—
393C4⋊C4—31no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—121no—
399C4⋊C4—121no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C42—61no—
406C42—61no—
407C42—61no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42D3×D411yescharacteristic
411C42D3×D411yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3—81no—
418C22×C3—81no—
419C22×C3C2×#12111yescharacteristic
420C22×C3#55211yescharacteristic
421C22×C3#55611yescharacteristic
422C2×D3—41no—
423C2×D3—41no—
424C2×D3—41no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—81no—
429C3×C4#47112yes—
430C3×C4#52511yescharacteristic
431C3×C4#53611yescharacteristic
432C3×C4#54711yescharacteristic
433C3×C4#54811yescharacteristic
434Dic3—41no—
435Dic3—41no—
436Dic3—41no—
437Dic3—82no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—61no—
442C23—81no—
443C23—81no—
444C23C2×#23511yescharacteristic
445Q8—61no—
446Q8—61no—
447Q8—242no—
448Q8—242no—
449D4—22no—
450D4—22no—
451D4—41no—
452D4—41no—
453D4—61no—
454D4—61no—
455D4—121no—
456D4—121no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—241no—
471C2×C4—242no—
472C2×C4#88411yescharacteristic
473C2×C4#88411yescharacteristic
474C2×C4#88411yescharacteristic
475C2×C4#88411yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4C2×#23711yescharacteristic
479C2×C4#100711yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4#101011yescharacteristic
482C8—42no—
483C8—42no—
484C8—62no—
485C8—62no—
486C8—62no—
487C8—121no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—82no—
491C2×C3#241511yescharacteristic
492C2×C3C2×#47111yescharacteristic
493C2×C3#306811yescharacteristic
494D3—81no—
495C22—21no—
496C22—21no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22—81no—
501C22—81no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22C2×#88411yescharacteristic
506C22#551311yescharacteristic
507C22#553311yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—81no—
512C4—121no—
513C4—121no—
514C4—121no—
515C4—242no—
516C4#474812yes—
517C4#559811yescharacteristic
518C4#560611yescharacteristic
519C4#562811yescharacteristic
520C4#563111yescharacteristic
521C3#2472111yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—82no—
525C2—241no—
526C2#7402011yescharacteristic
527C2C2×#474811yescharacteristic
528C2#7872011yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1800C2
socleC22×C3C2×#121
prime radicalC2×#932C22
hypercenter#1800D3
hypocenterC3#24721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. C2×#5008C2semidirect
3. #79060C2semidirect
4. #79040C2semidirect
5. C3×#1800C2semidirect
6. #70524C2semidirect
7. #70520C2semidirect
9. #70072C2semidirect
10. #69011C2semidirect
11. #68984C2semidirect
60. #1800D3semidirect
521. C3#24721semidirect

Maximal subgroups

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

subgroupindex
C2×#50082
C4×#9322
#790602
#790402
C3×#18002
#705242
#705202
#700922
#700722
#690112
#689842
#247213