Finite Group Databasebeta

← #10836664

Subgroups — #10836664

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
1#79060C211yesmaximal, characteristic
2C2×#4850C211yesmaximal, characteristic, complemented
3#79000C211yesmaximal, characteristic
4C4×#909C211yesmaximal, characteristic
5C3×#1950C211yesmaximal, characteristic, complemented
6#70716C212yesmaximal
7#70701C212yesmaximal, complemented
8#70528C211yesmaximal, characteristic, complemented
9#69879C211yesmaximal, characteristic, complemented
10#68930C212yesmaximal, complemented
11#68921C212yesmaximal
12#24722—31nomaximal, Sylow
13C2×#909C2211yescharacteristic
14C4×#235C2211yescharacteristic
15C2×#869C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×#108C2212yes—
18C3×#423C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×C4×M16C2211yescharacteristic
21C2×#824C2212yes—
22#5133C2211yescharacteristic
23#5133C2212yes—
24#5121C2211yescharacteristic
25#5121C2212yes—
26#5120C2211yescharacteristic
27#5110C2212yes—
28C2×#818C2212yes—
29C2×#816C2212yes—
30#5101C2212yes—
31#4924C2211yescharacteristic
32#4851—21no—
33#4850—21no—
34#4850—21no—
35#4850—21no—
36#4850—22no—
37#4846—21no—
38#4842—21no—
39#4840—22no—
40#4835—21no—
41#4833—21no—
42#4716—21no—
43#4710—21no—
44C4×Dic12—21no—
45C4×#204—21no—
46C3×#341C2212yes—
47#4554C2212yes—
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#1950D311yescharacteristic, complemented
59#1930—32no—
60#1801—31no—
61C2×#235C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6C2311yescharacteristic
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2312yes—
66C2×C3×M16C2311yescharacteristic
67C22×C3×C8C2312yes—
68C2×C3×C42C2311yescharacteristic
69C4×#67—21no—
70C4×#67—21no—
71#926C2311yescharacteristic
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×Dic3⋊C4—21no—
76C2×C4×Dic3—21no—
77C2×#208C2312yes—
78#909—22no—
79#909—22no—
80C2×Dic12—21no—
81C2×#204—21no—
82C4×D12D411yescharacteristic
83C4×D12D411yescharacteristic
84#872—21no—
85C42×D3—21no—
86#869—21no—
87#869—21no—
88#869—21no—
89#869—21no—
90C4×Dic6—21no—
91C4×Dic6—21no—
92C4×Dic6—21no—
93C4×Dic6—21no—
94C4×Dic6D411yescharacteristic
95C4×Dic6D411yescharacteristic
96C3×#108—24no—
97C3×C4⋊C8—21no—
98C3×C4⋊C8—21no—
99C3×C4⋊C8D411yescharacteristic
100C3×C4⋊C8D411yescharacteristic
101C3×C4⋊C8D412yes—
102C3×#97—21no—
103C3×C4×C8—21no—
104#824—24no—
105#821—21no—
106#821—22no—
107(C2×D3)⋊C8—22no—
108(C2×D3)⋊C8—22no—
109#818—22no—
110#818—22no—
111#818—21no—
112#818—22no—
113#817—21no—
114#817—22no—
115#816—21no—
116#816—22no—
117#816—22no—
118#816—22no—
119#816—21no—
120#816—21no—
121#815—22no—
122C8×Dic3—22no—
123C4⋊(C3⋊C8)—21no—
124C4⋊(C3⋊C8)—21no—
125C2×#137—31no—
126C2×C4⋊Q8—31no—
127C4×#41—31no—
128C2×C4×Q8—31no—
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×#108C2×D312yes—
144#428—32no—
145#428—31no—
146C2×#107—32no—
147#423—31no—
148#423C2×D311yescharacteristic
149C2×C4⋊C8C2×D311yescharacteristic
150#419—31no—
151#417—32no—
152#417—31no—
153C2×#103—32no—
154C2×#101—32no—
155#408—32no—
156#405—32no—
157C4×M16C2×D311yescharacteristic
158#343—32no—
159#343—32no—
160#343—32no—
161#341C2×D312yes—
162C22×C3×C4#4111yescharacteristic
163C22×C3×C4#4111yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169#235—21no—
170#235—21no—
171#235—21no—
172#235—21no—
173C2×D12C2×D411yescharacteristic
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—41no—
183C2×Dic6—41no—
184C2×Dic6—82no—
185C2×Dic6C2×D411yescharacteristic
186C2×Dic6#4112yes—
187C3×M16—21no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—21no—
191C3×M16—21no—
192C3×M16#4112yes—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8C2×D412yes—
199C2×C3×C8C2×D412yes—
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#217—21no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—21no—
209C4⋊Dic3—22no—
210C4⋊Dic3—21no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213C4⋊Dic3C2×D411yescharacteristic
214C4⋊Dic3C2×D411yescharacteristic
215Dic3⋊C4—21no—
216Dic3⋊C4—21no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219C4×Dic3—21no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—41no—
223C4×Dic3—41no—
224#208—22no—
225#208—22no—
226C2×C3⋊C8—22no—
227Dic12—24no—
228#204—24no—
229C2×#41—31no—
230C22×Q8—31no—
231C22×Q8—61no—
232#137—62no—
233#137—62no—
234C2×Q16—61no—
235C2×SD16—61no—
236C2×M16—32no—
237C2×M16C22×D311yescharacteristic
238C2×M16C22×D312yes—
239C2×M16C22×D311yescharacteristic
240C22×C8C22×D312yes—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—61no—
248C4×Q8—61no—
249C4×Q8—61no—
250C4×Q8—61no—
251C4×D4—31no—
252C4×D4—31no—
253C4×D4—61no—
254C4×D4—61no—
255#117—31no—
256#117—61no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C42—61no—
262C2×C42—61no—
263C2×C42C22×D311yescharacteristic
264#108—24no—
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—61no—
274C4⋊C8—61no—
275C4⋊C8D1211yescharacteristic
276C4⋊C8D1211yescharacteristic
277C4⋊C8D1212yes—
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—64no—
287C22⋊C8—62no—
288C22⋊C8—62no—
289#97—21no—
290#97—62no—
291C4×C8—21no—
292C4×C8—62no—
293C23×C3C2×#4111yescharacteristic
294C22×D3—21no—
295C2×C3×C4—22no—
296C2×C3×C4—41no—
297C2×C3×C4#12111yescharacteristic
298C2×C3×C4#12111yescharacteristic
299C2×C3×C4#12111yescharacteristic
300C2×C3×C4#12111yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C42−1+411yescharacteristic
306C2×C3×C42−1+411yescharacteristic
307#67—41no—
308#67—41no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—82no—
318C2×Dic3—81no—
319D12—21no—
320D12—21no—
321C4×D3—41no—
322C4×D3—41no—
323C4×D3—42no—
324C4×D3—81no—
325Dic6—22no—
326Dic6—21no—
327Dic6—21no—
328Dic6—22no—
329Dic6—41no—
330Dic6—41no—
331Dic6—82no—
332Dic6—82no—
333C3×C8—22no—
334C3×C8—22no—
335C3×C8—22no—
336C3×C8—42no—
337C3×C8—41no—
338C3⋊C8—42no—
339#41—61no—
340#41—61no—
341#41—61no—
342#41—61no—
343C2×Q8—31no—
344C2×Q8—32no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—121no—
351C2×Q8—121no—
352C2×Q8—242no—
353C2×D4—31no—
354C2×D4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4#23911yescharacteristic
362C22×C4#23911yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—64no—
365SD16—64no—
366M16—21no—
367M16—22no—
368M16—21no—
369M16—22no—
370M16—21no—
371M16—62no—
372M16—62no—
373M16#23712yes—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—62no—
380C2×C8C2×D1212yes—
381C2×C8C2×D1212yes—
382C4⋊C4—31no—
383C4⋊C4—31no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—121no—
391C4⋊C4—121no—
392C4⋊C4—121no—
393C4⋊C4—121no—
394C22⋊C4—61no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C42—61no—
398C42—61no—
399C42—61no—
400C42—121no—
401C42—121no—
402C42C2×D1211yescharacteristic
403C42C2×D1211yescharacteristic
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#47312yes—
418C3×C4#52511yescharacteristic
419C3×C4#53611yescharacteristic
420C3×C4#54711yescharacteristic
421C3×C4#54811yescharacteristic
422Dic3—42no—
423Dic3—41no—
424Dic3—41no—
425Dic3—41no—
426Dic3—82no—
427Dic3—81no—
428C23—61no—
429C23C2×#23911yescharacteristic
430Q8—62no—
431Q8—62no—
432Q8—61no—
433Q8—61no—
434Q8—121no—
435Q8—121no—
436Q8—242no—
437Q8—242no—
438D4—61no—
439D4—61no—
440D4—121no—
441D4—121no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—62no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—241no—
456C2×C4—242no—
457C2×C4—241no—
458C2×C4#89511yescharacteristic
459C2×C4#89511yescharacteristic
460C2×C4#89511yescharacteristic
461C2×C4#89511yescharacteristic
462C2×C4C22×D1211yescharacteristic
463C2×C4C2×D3×D411yescharacteristic
464C2×C4C2×#23711yescharacteristic
465C2×C4#100711yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C2×C4#101011yescharacteristic
468C8—22no—
469C8—22no—
470C8—22no—
471C8—42no—
472C8—41no—
473C8—122no—
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×#89511yescharacteristic
486C22#569411yescharacteristic
487C22#573211yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—121no—
492C4—121no—
493C4—121no—
494C4—122no—
495C4—242no—
496C4—241no—
497C4#489612yes—
498C4#555511yescharacteristic
499C4#557611yescharacteristic
500C4#560611yescharacteristic
501C4#562811yescharacteristic
502C3#2472211yescharacteristic, complemented, Sylow
503C2—21no—
504C2—21no—
505C2—241no—
506C2#7621011yescharacteristic
507C2C2×#489611yescharacteristic
508C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1950C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#1950D3
hypocenterC3#24722
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
2. C2×#4850C2semidirect
5. C3×#1950C2semidirect
7. #70701C2semidirect
8. #70528C2semidirect
9. #69879C2semidirect
10. #68930C2semidirect
58. #1950D3semidirect
502. C3#24722semidirect

Maximal subgroups

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

subgroupindex
#790602
C2×#48502
#790002
C4×#9092
C3×#19502
#707162
#707012
#705282
#698792
#689302
#689212
#247223