Finite Group Databasebeta

← #10950932

Subgroups — #10950932

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83742C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊C16C211yesmaximal, characteristic, complemented
4#71414C212yesmaximal, complemented
5#71413C212yesmaximal, complemented
6#69110C211yesmaximal, characteristic, complemented
7#69109C212yesmaximal, complemented
8#69108C211yesmaximal, characteristic, complemented
9#69099C211yesmaximal, characteristic, complemented
10#69097C212yesmaximal, complemented
11#69096C211yesmaximal, characteristic, complemented
12#33754—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14#5791—21no—
15C2×#930—21no—
16C2×#904C2211yescharacteristic, complemented
17C22×D24C2211yescharacteristic
18C22×C3×C16C2212yescomplemented
19#5116C2211yescharacteristic
20#5116C2211yescharacteristic, complemented
21#5115—21no—
22#5115—21no—
23#5115C2212yes—
24#5115C2212yes—
25#5115C2211yescharacteristic
26#5115C2211yescharacteristic, complemented
27#5114—21no—
28#5114—21no—
29#5113—21no—
30C2×#821—21no—
31#5106C2211yescharacteristic, complemented
32C2×#818C2211yescharacteristic
33C2×#800C2212yescomplemented
34C2×D48C2212yescomplemented
35C3×C22⋊C16C2211yescharacteristic
36C3×C22⋊C16C2211yescharacteristic
37C3×C22⋊C16C2212yescomplemented
38#4509C2212yescomplemented
39#4509C2212yescomplemented
40#4509C2212yescomplemented
41#4505C2212yescomplemented
42#4504C2212yescomplemented
43#4503C2212yescomplemented
44#3050—31no—
45C2×#466—31no—
46#2236—31no—
47#2234—32no—
48#2233—31no—
49#2217—32no—
50#2215—32no—
51#2187—31no—
52#2186—32no—
53#2184—31no—
54C2×C22⋊C16D311yescharacteristic, complemented
55C23×C3×C4D411yescharacteristic
56C22×#67—41no—
57C2×#235—21no—
58C22×D12—21no—
59C22×C3×C8—42no—
60C22×C3×C8D411yescharacteristic
61C22×C3×C8D411yescharacteristic
62C22×C3×C8D411yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66#953—41no—
67#930—22no—
68#930—22no—
69#930—21no—
70#930—21no—
71#930—41no—
72#930—41no—
73#929—41no—
74C4×#67—41no—
75C4×#67—41no—
76C2×(C2×D3)⋊C4—41no—
77#926—21no—
78C2×C4⋊Dic3—21no—
79#924—21no—
80#924—21no—
81C2×Dic12C2311yescharacteristic
82#904—42no—
83C2×D24—21no—
84C2×D24—42no—
85C2×D24C2311yescharacteristic
86C2×D24C2312yes—
87C2×#204—21no—
88C2×C3×C16—42no—
89C2×C3×C16C2312yes—
90C2×C3×C16C2312yes—
91#821—22no—
92#821—22no—
93#821—21no—
94#821—21no—
95#818—21no—
96#818C2312yes—
97#818C2311yescharacteristic
98#818C2311yescharacteristic
99#817—21no—
100#816—21no—
101#816—21no—
102#800—42no—
103D48—42no—
104C2×#135—31no—
105C22×D8—31no—
106C23×C8C2×D311yescharacteristic
107#525—61no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×D16—32no—
111C22×C16C2×D312yescomplemented
112#467—31no—
113#467—31no—
114#466—31no—
115#466—32no—
116#466—31no—
117#466—32no—
118#466—61no—
119#466—61no—
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107—31no—
124#416—61no—
125C2×#102—61no—
126#367—32no—
127#366—32no—
128#358—32no—
129#357—32no—
130#357—32no—
131#357—32no—
132C22⋊C16C2×D311yescharacteristic
133C22⋊C16C2×D311yescharacteristic
134C22⋊C16C2×D312yescomplemented
135C24×C3D811yescharacteristic
136C23×D3—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4D811yescharacteristic
140C22×C3×C4D811yescharacteristic
141C22×C3×C4D811yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—42no—
149C2×#67—42no—
150C2×#67—84no—
151C22×Dic3—41no—
152#235—42no—
153#235—42no—
154C2×D12—21no—
155C2×D12—22no—
156C2×D12—42no—
157C2×D12—42no—
158C2×D12—41no—
159C2×C4×D3—41no—
160C2×Dic6—21no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—41no—
165C2×C3×C8—41no—
166C2×C3×C8—41no—
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2411yescharacteristic
171#217—41no—
172#217—41no—
173#217—41no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—42no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—41no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3—22no—
181C4⋊Dic3—41no—
182Dic3⋊C4—41no—
183Dic3⋊C4—41no—
184C4×Dic3—41no—
185Dic12—22no—
186D24—22no—
187D24—22no—
188D24—22no—
189D24—42no—
190#204—42no—
191C3×C16—22no—
192C3×C16—22no—
193C2×#41—61no—
194C22×D4—61no—
195C22×D4—121no—
196C23×C4D1211yescharacteristic
197#135—122no—
198C2×Q16—31no—
199C2×SD16—61no—
200C2×D8—32no—
201C2×D8—31no—
202C2×D8—61no—
203C2×D8—122no—
204C22×C8—42no—
205C22×C8D1211yescharacteristic
206C22×C8D1211yescharacteristic
207C22×C8D1211yescharacteristic
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211#123—121no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214#121—62no—
215#121—62no—
216#121—61no—
217#121—61no—
218#121—121no—
219#121—121no—
220#120—121no—
221C4×D4—121no—
222C4×D4—121no—
223#117—61no—
224C2×C4⋊C4—61no—
225C2×C22⋊C4—121no—
226SD32—122no—
227D16—122no—
228C2×C16—42no—
229C2×C16C22×D312yes—
230C2×C16C22×D312yes—
231#107—31no—
232#107—31no—
233#107—32no—
234#107—61no—
235#106—61no—
236#103—61no—
237#103—61no—
238#102—62no—
239#102—62no—
240#102—61no—
241#102—61no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3C2×D811yescharacteristic
246C23×C3C2×D811yescharacteristic
247C23×C3C2×D811yescharacteristic
248C22×D3—41no—
249C22×D3—42no—
250C22×D3—82no—
251C22×D3—82no—
252C22×D3—82no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4C2×D811yescharacteristic
263C2×C3×C4C2×D811yescharacteristic
264C2×C3×C4C2×D811yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266#67—82no—
267#67—84no—
268#67—82no—
269#67—82no—
270#67—84no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—82no—
276D12—41no—
277D12—42no—
278D12—42no—
279D12—41no—
280D12—82no—
281C4×D3—82no—
282Dic6—41no—
283Dic6—41no—
284C3×C8—21no—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8H(5,𝔽2)12yes—
288C24—121no—
289C24D2411yescharacteristic
290#41—122no—
291#41—122no—
292C2×Q8—61no—
293C2×D4—62no—
294C2×D4—61no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—122no—
301C2×D4—122no—
302C2×D4—121no—
303C2×D4—244no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4D2411yescharacteristic
309C22×C4D2411yescharacteristic
310C22×C4D2411yescharacteristic
311C22×C4C2×D1211yescharacteristic
312C22×C4C2×D1211yescharacteristic
313C22×C4C2×D1211yescharacteristic
314Q16—62no—
315SD16—122no—
316D8—62no—
317D8—62no—
318D8—62no—
319D8—122no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×D1211yescharacteristic
327C2×C8C2×D1211yescharacteristic
328C2×C8C2×D1211yescharacteristic
329C2×C8C23×D311yescharacteristic
330C4⋊C4—62no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—121no—
334C4⋊C4—121no—
335C4⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—122no—
340C22⋊C4—122no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—121no—
344C16—22no—
345C16—22no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—44no—
354C22×C3#50812yes—
355C22×C3#50912yes—
356C22×C3C22×D811yescharacteristic
357C2×D3—81no—
358C2×D3—82no—
359C2×D3—82no—
360C2×D3—81no—
361C2×D3—81no—
362C2×D3—82no—
363C2×D3—164no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4#53411yescharacteristic
368C3×C4#57511yescharacteristic
369Dic3—81no—
370Dic3—82no—
371Dic3—81no—
372Dic3—81no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—121no—
377C23—122no—
378C23—242no—
379C23—242no—
380C23—242no—
381C23C2×D2411yescharacteristic
382C23C2×D2411yescharacteristic
383C23C2×D2411yescharacteristic
384Q8—121no—
385Q8—121no—
386D4—122no—
387D4—121no—
388D4—122no—
389D4—121no—
390D4—242no—
391D4—244no—
392D4—242no—
393D4—242no—
394D4—244no—
395D4—242no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—242no—
410C2×C4—242no—
411C2×C4C2×D2411yescharacteristic
412C2×C4C2×D2411yescharacteristic
413C2×C4C2×D2411yescharacteristic
414C2×C4C22×D1211yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8#100412yes—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—41no—
422C2×C3#318611yescharacteristic
423C2×C3C2×#50811yescharacteristic
424C2×C3C2×#50911yescharacteristic
425D3—161no—
426D3—162no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22—241no—
436C22—242no—
437C22—241no—
438C22—242no—
439C22—242no—
440C22—241no—
441C22—484no—
442C22#490412yes—
443C22#490812yes—
444C22C22×D2411yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—241no—
449C4—241no—
450C4—242no—
451C4—241no—
452C4#549311yescharacteristic
453C4#574811yescharacteristic
454C3#3375411yescharacteristic, complemented, Sylow
455C2—22no—
456C2—22no—
457C2—41no—
458C2—482no—
459C2—481no—
460C2#7861111yescharacteristic
461C2C2×#490411yescharacteristic
462C2C2×#490811yescharacteristic
subgroupG/H
centerC22C22×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×C22⋊C16C2
socleC22×C3C22×D8
hypercenterC2×C22⋊C16D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
1. #83742C2semidirect
2. C2×#5115C2semidirect
3. C2×C3×C22⋊C16C2semidirect
4. #71414C2semidirect
5. #71413C2semidirect
6. #69110C2semidirect
7. #69109C2semidirect
8. #69108C2semidirect
9. #69099C2semidirect
10. #69097C2semidirect
11. #69096C2semidirect
16. C2×#904C22semidirect
18. C22×C3×C16C22semidirect
20. #5116C22semidirect
26. #5115C22semidirect
31. #5106C22semidirect
33. C2×#800C22semidirect
34. C2×D48C22semidirect
37. C3×C22⋊C16C22semidirect
38. #4509C22semidirect
39. #4509C22semidirect
40. #4509C22semidirect
41. #4505C22semidirect
42. #4504C22semidirect
43. #4503C22semidirect
54. C2×C22⋊C16D3semidirect
111. C22×C16C2×D3semidirect
134. C22⋊C16C2×D3semidirect
454. C3#33754semidirect

Maximal subgroups

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

subgroupindex
#837422
C2×#51152
C2×C3×C22⋊C162
#714142
#714132
#691102
#691092
#691082
#690992
#690972
#690962
#337543