Finite Group Databasebeta

← #10690691

Subgroups — #10690691

Order 768.

630 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#79286C211yesmaximal, characteristic, complemented
2#73389C211yesmaximal, characteristic
3#70783C211yesmaximal, characteristic, complemented
4#70757C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic, complemented
6C3×#1274C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9101—31nomaximal, Sylow
9C2×#874C411yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#4939C411yescharacteristic
16#4939C411yescharacteristic
17#4936C411yescharacteristic
18#4925C2211yescharacteristic
19#4925C2211yescharacteristic
20#4834—22no—
21D3×C22⋊C8—22no—
22C8×D12—22no—
23#4665—22no—
24#4665—22no—
25#4483—22no—
26#4468—22no—
27#4459—22no—
28#2972—31no—
29#2328—31no—
30#1478—31no—
31#1478—31no—
32#1460—31no—
33#1274—31no—
34#1274D311yescharacteristic, complemented
35C22×D12—22no—
36C22×D12C2×C411yescharacteristic
37C22×D12C2×C411yescharacteristic
38C22×C4×D3—21no—
39C22×C4×D3D412yes—
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4D412yes—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×D3×C8—24no—
50#874—22no—
51#874C814yescomplemented
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C4×D12—21no—
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×D4—31no—
79C8×D4—62no—
80C8×D4—62no—
81#431—31no—
82#431—31no—
83#431C2×D311yescharacteristic
84#431C2×D311yescharacteristic
85C2×C22⋊C8—62no—
86#390—31no—
87#390—31no—
88#386—62no—
89#386—62no—
90#382—31no—
91#376C2×D311yescharacteristic
92D4⋊C8—62no—
93D4⋊C8—62no—
94#323—62no—
95C23×D3—21no—
96C23×D3C22⋊C412yes—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×Dic3—21no—
103C22×Dic3C22⋊C412yes—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—28no—
114C2×D12—82no—
115C2×D12—82no—
116C2×D12C2×C812yes—
117C2×D12C2×C812yescomplemented
118C2×D12M1612yes—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—82no—
124C2×C4×D3—82no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C42—21no—
128C3×C42—21no—
129C3×C42—21no—
130C3×C42C2×C812yescomplemented
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—82no—
140(C2×D3)⋊C4—82no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147D3×C8—84no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—62no—
151C23×C4—32no—
152C23×C4—61no—
153C22×C8—64no—
154#127—34no—
155#127—62no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4—61no—
162C4×D4—61no—
163C4×D4—61no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C22⋊C4—32no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C42C4×D311yescharacteristic
171C2×C42C4×D311yescharacteristic
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×C8D1212yescomplemented
186#95—21no—
187#95—21no—
188#95—62no—
189C23×C3C2×C22⋊C411yescharacteristic
190C22×D3—22no—
191C22×D3—22no—
192C22×D3—22no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—42no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3C22⋊C814yes—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4C22×C811yescharacteristic
215C2×C3×C4C2×M1611yescharacteristic
216C2×C3×C4#13111yescharacteristic
217C2×C3×C4#13111yescharacteristic
218C2×Dic3—22no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222D12—22no—
223D12—22no—
224D12—22no—
225D12—22no—
226D12—22no—
227D12—22no—
228D12—41no—
229D12—41no—
230D12—88no—
231C4×D3—44no—
232C4×D3—82no—
233C4×D3—82no—
234C4×D3—82no—
235C3×C8—42no—
236C3×C8—81no—
237C3⋊C8—42no—
238C3⋊C8—81no—
239C24—32no—
240C24—61no—
241C2×D4—32no—
242C2×D4—32no—
243C2×D4—32no—
244C2×D4—68no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—242no—
255C2×D4—242no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—32no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—61no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—242no—
265C22×C4—242no—
266C22×C4C2×C4×D311yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—62no—
272C2×C8—121no—
273C2×C8—244no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—242no—
283C22⋊C4—242no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42D3×C812yescomplemented
288C42C2×D1211yescharacteristic
289C42#23511yescharacteristic
290C42D3×D411yescharacteristic
291C42#23911yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3#38611yescharacteristic
295C22×C3C2×C22⋊C811yescharacteristic
296C22×C3#40811yescharacteristic
297C2×D3—24no—
298C2×D3—42no—
299C2×D3—42no—
300C2×D3—44no—
301C2×D3—44no—
302C2×D3—42no—
303C2×D3—82no—
304C2×D3—82no—
305C2×D3—81no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4—81no—
313C3×C4C8×D411yescharacteristic
314C3×C4C8×D411yescharacteristic
315C3×C4#43511yescharacteristic
316C3×C4#43511yescharacteristic
317Dic3—42no—
318Dic3—81no—
319C23—34no—
320C23—62no—
321C23—62no—
322C23—62no—
323C23—122no—
324C23—122no—
325C23—122no—
326C23—122no—
327C23—121no—
328C23—121no—
329C23D3×C22⋊C411yescharacteristic
330D4—62no—
331D4—62no—
332D4—62no—
333D4—62no—
334D4—62no—
335D4—62no—
336D4—121no—
337D4—121no—
338D4—248no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—62no—
348C2×C4—124no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—242no—
353C2×C4—242no—
354C2×C4—242no—
355C2×C4C4×D1211yescharacteristic
356C2×C4#88211yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#89311yescharacteristic
359C2×C4#89411yescharacteristic
360C2×C4#89511yescharacteristic
361C2×C4C2×D3×C811yescharacteristic
362C2×C4#90111yescharacteristic
363C2×C4D3×M1611yescharacteristic
364C2×C4#90711yescharacteristic
365C8—42no—
366C8—81no—
367C8—122no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3#186911yescharacteristic
372C2×C3#191011yescharacteristic
373C2×C3#191111yescharacteristic
374D3—44no—
375D3—82no—
376C22—21no—
377C22—21no—
378C22—64no—
379C22—124no—
380C22—124no—
381C22—122no—
382C22—122no—
383C22—122no—
384C22—242no—
385C22—242no—
386C22—241no—
387C22—241no—
388C22#466511yescharacteristic
389C22D3×C22⋊C811yescharacteristic
390C22#472711yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—81no—
397C4—122no—
398C4—241no—
399C4C8×D1211yescharacteristic
400C4#470611yescharacteristic
401C4#483411yescharacteristic
402C4#483611yescharacteristic
403C3#910111yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—124no—
407C2—242no—
408C2#6866811yescharacteristic
409C2#6870511yescharacteristic
410C2#6871311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#386
prime radicalC2×#874C4
cosocleC2×C3×C42C23
hypercenter#1274D3
hypocenterC3#9101
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #79286C2semidirect
3. #70783C2semidirect
4. #70757C2semidirect
5. #70756C2semidirect
6. C3×#1274C2semidirect
7. #67614C2semidirect
34. #1274D3semidirect
51. #874C8semidirect
65. C3×C4×C8D4semidirect
75. C4×C3⋊C8D4semidirect
117. C2×D12C2×C8semidirect
130. C3×C42C2×C8semidirect
185. C4×C8D12semidirect
287. C42D3×C8semidirect
403. C3#9101semidirect

Maximal subgroups

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

subgroupindex
#792862
#733892
#707832
#707572
#707562
C3×#12742
#676142
#91013