Finite Group Databasebeta

← #10950942

Subgroups — #10950942

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3050C211yesmaximal, characteristic
2C2×#5116C211yesmaximal, characteristic, complemented
3#72083C212yesmaximal, complemented
4C2×#4549C211yesmaximal, characteristic, complemented
5#71087C211yesmaximal, characteristic, complemented
6#71081C212yesmaximal, complemented
7#71078C211yesmaximal, characteristic, complemented
8#71077C211yesmaximal, characteristic, complemented
9#71072C212yesmaximal, complemented
10#71068C211yesmaximal, characteristic, complemented
11#70655C212yesmaximal, complemented
12#33756—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C3×#525—21no—
16C2×#924—21no—
17C22×Dic12C2211yescharacteristic
18C3×#467C2211yescharacteristic
19C3×#467C2211yescharacteristic
20C3×#466C2211yescharacteristic
21C3×#466C2211yescharacteristic
22C3×#465—21no—
23C3×#465—21no—
24C3×#427C2211yescharacteristic
25C3×#416—21no—
26C2×#829C2212yescomplemented
27C2×#827C2212yescomplemented
28#5116—21no—
29#5116—21no—
30#5116C2212yes—
31#5116C2212yes—
32C2×#818C2211yescharacteristic
33C2×#816—21no—
34C22×C3⋊C16C2212yes—
35#4564C2212yescomplemented
36#4562C2212yescomplemented
37#4549C2212yes—
38#4549C2211yescharacteristic
39#4549C2211yescharacteristic
40#4492C2212yescomplemented
41#4492C2212yescomplemented
42#4490C2212yescomplemented
43#4489C2212yescomplemented
44#3050D311yescharacteristic
45C2×#467—31no—
46#2237—32no—
47#2235—31no—
48#2232—31no—
49#2217—32no—
50#2216—32no—
51#2189—32no—
52#2188—31no—
53#2185—31no—
54C2×C22⋊C16—31no—
55C2×C3×#41—21no—
56C23×C3×C4D411yescharacteristic
57C22×Dic6—21no—
58C3×#135—42no—
59C2×C3×Q16C2311yescharacteristic
60C2×C3×SD16—21no—
61C2×C3×D8C2311yescharacteristic
62C22×C3×C8—42no—
63C22×C3×C8D411yescharacteristic
64C22×C3×C8D411yescharacteristic
65C22×C3×C8D411yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×#123—41no—
70C3×C22⋊Q8—21no—
71C3×C22⋊Q8—21no—
72C3×#121—21no—
73C3×#121—21no—
74C3×#120—41no—
75C3×C4×D4—41no—
76C3×C4×D4—41no—
77C3×#117—21no—
78C2×#217—41no—
79C2×C4⋊Dic3—21no—
80#924—22no—
81#924—22no—
82#924—41no—
83#924—41no—
84C2×Dic3⋊C4—41no—
85C2×Dic12—21no—
86C2×Dic12—42no—
87C2×Dic12C2312yes—
88C3×#107C2311yescharacteristic
89C3×#107C2311yescharacteristic
90C3×#106—21no—
91C3×#103—21no—
92C3×#103—21no—
93C3×#102—21no—
94C3×#102—21no—
95#829—42no—
96#827—42no—
97#818—21no—
98#818C2312yes—
99#816—22no—
100#816—22no—
101C2×C3⋊C16—42no—
102C2×C3⋊C16C2312yes—
103C2×C3⋊C16C2312yes—
104C2×#135C2×D311yescharacteristic
105C22×Q16—31no—
106C23×C8C2×D311yescharacteristic
107#525—21no—
108C2×C22⋊Q8—61no—
109C2×Q32—32no—
110C2×SD32—32no—
111C22×C16—32no—
112#467—32no—
113#467—32no—
114#467—61no—
115#467—61no—
116#467C2×D311yescharacteristic
117#467C2×D311yescharacteristic
118#466C2×D311yescharacteristic
119#466C2×D311yescharacteristic
120#465—21no—
121#465—21no—
122#427C2×D311yescharacteristic
123C2×#107—31no—
124#416—21no—
125C2×#103—61no—
126#367—32no—
127#366—32no—
128#358—32no—
129#358—32no—
130#358—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—31no—
134C22⋊C16—32no—
135C24×C3D811yescharacteristic
136C3×#41—42no—
137C3×#41—42no—
138C2×C3×Q8—21no—
139C2×C3×D4—21no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4D811yescharacteristic
147C22×C3×C4D811yescharacteristic
148C22×C3×C4D811yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×Dic3—41no—
153C22×Dic3—41no—
154C2×Dic6—22no—
155C2×Dic6—42no—
156C2×Dic6—42no—
157C2×Dic6—41no—
158C3×Q16—22no—
159C3×SD16—42no—
160C3×D8—22no—
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
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—41no—
174C3×C4⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C42—41no—
181#217—42no—
182#217—42no—
183C4⋊Dic3—22no—
184C4⋊Dic3—41no—
185Dic3⋊C4—42no—
186Dic3⋊C4—42no—
187Dic12—22no—
188Dic12—22no—
189Dic12—42no—
190C3⋊C16—22no—
191C3⋊C16—22no—
192C2×#41—21no—
193C22×Q8—61no—
194C23×C4#6711yescharacteristic
195#135—42no—
196C2×Q16—32no—
197C2×Q16—61no—
198C2×Q16—122no—
199C2×Q16C22×D311yescharacteristic
200C2×SD16—21no—
201C2×D8C22×D311yescharacteristic
202C22×C8—42no—
203C22×C8#6711yescharacteristic
204C22×C8#6711yescharacteristic
205C22×C8#6711yescharacteristic
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209#123—41no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—62no—
213C22⋊Q8—62no—
214C22⋊Q8—121no—
215C22⋊Q8—121no—
216#121—21no—
217#121—21no—
218#120—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—21no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225Q32—122no—
226SD32—122no—
227C2×C16—32no—
228C2×C16—32no—
229C2×C16—122no—
230#107—32no—
231#107—61no—
232#107C22×D311yescharacteristic
233#107C22×D311yescharacteristic
234#106—21no—
235#103—21no—
236#103—21no—
237#103—62no—
238#103—62no—
239#102—21no—
240#102—21no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3C2×D811yescharacteristic
246C23×C3C2×D811yescharacteristic
247C23×C3C2×D811yescharacteristic
248C3×Q8—41no—
249C3×Q8—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—82no—
253C3×D4—82no—
254C3×D4—82no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—82no—
268C2×C3×C4C2×D811yescharacteristic
269C2×C3×C4C2×D811yescharacteristic
270C2×C3×C4C2×D811yescharacteristic
271C2×C3×C4C22×D411yescharacteristic
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—82no—
275C2×Dic3—82no—
276Dic6—42no—
277Dic6—42no—
278Dic6—82no—
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—21no—
282C3×C8H(5,𝔽2)12yes—
283C24#21311yescharacteristic
284#41—42no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—62no—
288C2×Q8—122no—
289C2×Q8—122no—
290C2×Q8—121no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4#21311yescharacteristic
301C22×C4#21311yescharacteristic
302C22×C4#21311yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306Q16—22no—
307Q16—62no—
308Q16—62no—
309Q16—122no—
310SD16—42no—
311D8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8C2×#6711yescharacteristic
319C2×C8C2×#6711yescharacteristic
320C2×C8C2×#6711yescharacteristic
321C2×C8C23×D311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—62no—
327C4⋊C4—121no—
328C4⋊C4—122no—
329C4⋊C4—122no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—122no—
336C22⋊C4—122no—
337C42—41no—
338C16—62no—
339C16—62no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#50812yes—
352C22×C3#51012yes—
353C22×C3C22×D811yescharacteristic
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—81no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4#53411yescharacteristic
361C3×C4#57511yescharacteristic
362Dic3—82no—
363Dic3—82no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23C2×#21311yescharacteristic
369C23C2×#21311yescharacteristic
370C23C2×#21311yescharacteristic
371Q8—41no—
372Q8—41no—
373Q8—122no—
374Q8—122no—
375Q8—242no—
376D4—41no—
377D4—41no—
378D4—82no—
379D4—82no—
380D4—82no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—242no—
397C2×C4—242no—
398C2×C4C2×#21311yescharacteristic
399C2×C4C2×#21311yescharacteristic
400C2×C4C2×#21311yescharacteristic
401C2×C4C22×#6711yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8#100412yes—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3—161no—
410C2×C3#318611yescharacteristic
411C2×C3C2×#50811yescharacteristic
412C2×C3C2×#51011yescharacteristic
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#519412yes—
425C22#519512yes—
426C22C22×#21311yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—242no—
434C4—242no—
435C4#580411yescharacteristic
436C4#582211yescharacteristic
437C3#3375611yescharacteristic, complemented, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—161no—
442C2#8262511yescharacteristic
443C2C2×#519411yescharacteristic
444C2C2×#519511yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
Fitting, prime radicalC3×#3050C2
socleC22×C3C22×D8
hypercenter#3050D3
hypocenterC3#33756
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
2. C2×#5116C2semidirect
3. #72083C2semidirect
4. C2×#4549C2semidirect
5. #71087C2semidirect
6. #71081C2semidirect
7. #71078C2semidirect
8. #71077C2semidirect
9. #71072C2semidirect
10. #71068C2semidirect
11. #70655C2semidirect
26. C2×#829C22semidirect
27. C2×#827C22semidirect
35. #4564C22semidirect
36. #4562C22semidirect
40. #4492C22semidirect
41. #4492C22semidirect
42. #4490C22semidirect
43. #4489C22semidirect
437. C3#33756semidirect

Maximal subgroups

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

subgroupindex
C3×#30502
C2×#51162
#720832
C2×#45492
#710872
#710812
#710782
#710772
#710722
#710682
#706552
#337563