Finite Group Databasebeta

← #10690709

Subgroups — #10690709

Order 768.

602 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82329C211yesmaximal, characteristic
2#79288C211yesmaximal, characteristic, complemented
3C3×#1478C211yesmaximal, characteristic, complemented
4#71597C211yesmaximal, characteristic, complemented
5#70783C211yesmaximal, characteristic, complemented
6#67737C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9105—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#940C411yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5221C411yescharacteristic
15#5213—22no—
16#4988C411yescharacteristic
17#4977C411yescharacteristic
18#4956—22no—
19#4932C2211yescharacteristic
20#4925C2211yescharacteristic
21#4925C2211yescharacteristic
22#4838—22no—
23#4726—22no—
24#4688—22no—
25C3×D4⋊C8—22no—
26D4⋊(C3⋊C8)—22no—
27#4468—22no—
28#2975—31no—
29#2328—31no—
30#1478—31no—
31#1478D311yescharacteristic, complemented
32#1460—31no—
33#1274—31no—
34#1274—31no—
35C22×C3×D4C2×C411yescharacteristic
36C23×C3×C4—21no—
37C22×#67—22no—
38C22×D12C2×C411yescharacteristic
39C22×C4×D3D412yes—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#217—22no—
48#940—22no—
49#940—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4D412yes—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×ΓL(1,𝔽25)—24no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3×C4×C8D412yescomplemented
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69#830—21no—
70#830—21no—
71(C2×D3)⋊C8—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yescomplemented
76C2×#127—31no—
77C2×C4×D4—31no—
78C2×C4×D4C2×D311yescharacteristic
79#436—62no—
80#436—62no—
81#431—31no—
82#431—31no—
83#431C2×D311yescharacteristic
84#431C2×D311yescharacteristic
85#407—62no—
86#390—31no—
87#390—31no—
88#386—62no—
89#386—62no—
90#382—31no—
91#376—31no—
92D4⋊C8—22no—
93D4⋊C8—62no—
94#323—62no—
95C24×C3—21no—
96C23×D3C22⋊C412yes—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4M1612yescomplemented
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—82no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C2×#67—28no—
111C2×#67—82no—
112C2×#67—82no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3C22⋊C412yes—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—22no—
123C2×C4×D3—22no—
124C2×C4×D3—82no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135#217—82no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4×Dic3—21no—
142C4×Dic3—21no—
143C4×Dic3—21no—
144C4×Dic3M1612yescomplemented
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147ΓL(1,𝔽25)—84no—
148C22×D4—31no—
149C22×D4—62no—
150C22×D4C4×D311yescharacteristic
151C23×C4—21no—
152C23×C4—32no—
153C2×M16—64no—
154#127—62no—
155#127—62no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—61no—
161C4×D4—61no—
162C4×D4—61no—
163C4×D4—61no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4C4×D311yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—32no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C42—31no—
171C2×C42—31no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—64no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182C4×C8—21no—
183C4×C8—32no—
184C4×C8—61no—
185C4×C8#6712yescomplemented
186#95—62no—
187#95—61no—
188#95—61no—
189C23×C3—22no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3C2×C22⋊C411yescharacteristic
195C22×D3—22no—
196C22×D3—22no—
197C22×D3—22no—
198C22×D3—42no—
199C22×D3—42no—
200C3×D4—22no—
201C3×D4—22no—
202C3×D4—41no—
203C3×D4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—82no—
209C2×C3×C4—82no—
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4C2×M1611yescharacteristic
217C2×C3×C4C2×M1611yescharacteristic
218C2×C3×C4#13111yescharacteristic
219C2×C3×C4#13111yescharacteristic
220#67—88no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229D12—41no—
230D12—41no—
231D12—41no—
232D12—41no—
233C4×D3—44no—
234C4×D3—82no—
235C3×C8—42no—
236C3×C8—81no—
237C3⋊C8—42no—
238C3⋊C8—81no—
239C24—21no—
240C24—32no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—68no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—242no—
254C2×D4—242no—
255C2×D4ΓL(1,𝔽25)12yescomplemented
256C22×C4—21no—
257C22×C4—32no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—61no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—61no—
264C22×C4—82no—
265C22×C4—242no—
266C22×C4C2×C4×D311yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269M16—244no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—62no—
273C2×C8—121no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—242no—
283C22⋊C4—242no—
284C42—32no—
285C42—61no—
286C42—61no—
287C42—61no—
288C42#23511yescharacteristic
289C42D3×D411yescharacteristic
290C42#23711yescharacteristic
291C42C2×#6711yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—44no—
295C22×C3—42no—
296C22×C3—82no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3#38611yescharacteristic
300C22×C3#40711yescharacteristic
301C22×C3#40811yescharacteristic
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—82no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—81no—
312C3×C4#43511yescharacteristic
313C3×C4#43511yescharacteristic
314C3×C4#43611yescharacteristic
315C3×C4#43611yescharacteristic
316Dic3—42no—
317Dic3—42no—
318Dic3—81no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23—62no—
325C23—62no—
326C23—62no—
327C23—122no—
328C23—122no—
329C23D3×C22⋊C411yescharacteristic
330D4—22no—
331D4—22no—
332D4—41no—
333D4—41no—
334D4—121no—
335D4—121no—
336D4—121no—
337D4—121no—
338D4—248no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—124no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—242no—
355C2×C4#67⋊C411yescharacteristic
356C2×C4#88311yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
359C2×C4#90111yescharacteristic
360C2×C4D3×M1611yescharacteristic
361C2×C4#90711yescharacteristic
362C2×C4C4×#6711yescharacteristic
363C2×C4#93711yescharacteristic
364C2×C4#93811yescharacteristic
365C8—42no—
366C8—81no—
367C8—122no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—82no—
372C2×C3#187011yescharacteristic
373C2×C3#191011yescharacteristic
374C2×C3#191111yescharacteristic
375D3—82no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—42no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22—124no—
384C22—122no—
385C22—122no—
386C22—122no—
387C22—242no—
388C22#472611yescharacteristic
389C22#472711yescharacteristic
390C22#495611yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—122no—
397C4—122no—
398C4—241no—
399C4#472911yescharacteristic
400C4#473011yescharacteristic
401C4#511111yescharacteristic
402C4#512811yescharacteristic
403C3#910511yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—82no—
407C2—242no—
408C2#6995811yescharacteristic
409C2#7033011yescharacteristic
410C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1478C2
socleC22×C3#386
prime radicalC2×#940C4
cosocleC2×C3×C42C23
hypercenter#1478D3
hypocenterC3#9105
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #79288C2semidirect
3. C3×#1478C2semidirect
4. #71597C2semidirect
5. #70783C2semidirect
6. #67737C2semidirect
7. #67737C2semidirect
31. #1478D3semidirect
65. C3×C4×C8D4semidirect
75. C4×C3⋊C8D4semidirect
102. C2×C3×D4M16semidirect
144. C4×Dic3M16semidirect
185. C4×C8#67semidirect
255. C2×D4ΓL(1,𝔽25)semidirect
403. C3#9105semidirect

Maximal subgroups

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

subgroupindex
#823292
#792882
C3×#14782
#715972
#707832
#677372
#91053