Finite Group Databasebeta

← #10690639

Subgroups — #10690639

Order 768.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79288C211yesmaximal, characteristic, complemented
2C2×#4711C211yesmaximal, characteristic
3#70783C211yesmaximal, characteristic, complemented
4#70757C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic, complemented
6C3×#1281C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9093—31nomaximal, Sylow
9C22×D24C411yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×#97C2211yescharacteristic
15C2×#821C411yescharacteristic
16C2×#821C411yescharacteristic
17C2×#817C411yescharacteristic
18#4925C2211yescharacteristic
19#4925C2211yescharacteristic
20#4838—22no—
21#4726—22no—
22#4711—22no—
23#4711—22no—
24#4688—22no—
25#4483—22no—
26#4468—22no—
27#4459—22no—
28#2975—31no—
29C2×#442—31no—
30#1478—31no—
31#1478—31no—
32#1460—31no—
33#1281D311yescharacteristic, complemented
34#1275—31no—
35C22×D12C2×C411yescharacteristic
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C22×C4×D3—21no—
39C22×C3×C8C2×C411yescharacteristic
40C22×C3×C8C2×C411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×D24—22no—
47C2×D24—22no—
48C2×D24—22no—
49C2×ΓL(1,𝔽25)—24no—
50C4×D12—21no—
51C4×D12—21no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×D12D412yes—
57C4×D12D412yes—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8D412yes—
62C3×#97—21no—
63C3×#97—21no—
64C3×C4×C8—21no—
65C3×C4×C8—21no—
66#830—21no—
67#830—21no—
68#821—22no—
69#821—22no—
70(C2×D3)⋊C8—24no—
71#817—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C22×D8—31no—
77C2×C4×D4—31no—
78C2×C4×D4—31no—
79#442—62no—
80#442—62no—
81#436—62no—
82#436—62no—
83#431—31no—
84#431—31no—
85#431C2×D311yescharacteristic
86#431C2×D311yescharacteristic
87C2×#106—31no—
88C2×#102—31no—
89C2×#102—31no—
90#407—62no—
91C2×#97C2×D311yescharacteristic
92D4⋊C8—62no—
93D4⋊C8—62no—
94#323—62no—
95C23×D3—21no—
96C23×D3—21no—
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×D12—21no—
103C2×D12—21no—
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×D12C22⋊C412yes—
113C2×D12M1612yescomplemented
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—41no—
117C2×C4×D3—41no—
118C2×C4×D3—82no—
119C2×C4×D3—82no—
120C2×C3×C8—21no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—22no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—41no—
127C2×C3×C8M1612yescomplemented
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3C22⋊C412yes—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142D24—28no—
143ΓL(1,𝔽25)—84no—
144C22×D4—31no—
145C22×D4—31no—
146C23×C4—61no—
147C23×C4—61no—
148C2×D8—62no—
149C2×D8—62no—
150C2×D8—62no—
151C2×M16—64no—
152C22×C8C4×D311yescharacteristic
153C22×C8C4×D311yescharacteristic
154C4×D4—32no—
155C4×D4—32no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#106—62no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#102—62no—
173#102—62no—
174C22⋊C8—21no—
175C22⋊C8—64no—
176C22⋊C8—61no—
177C22⋊C8—61no—
178C22⋊C8D1212yes—
179#97—21no—
180#97—21no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—32no—
184C4×C8—61no—
185C23×C3C4×D411yescharacteristic
186C22×D3—22no—
187C22×D3—22no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3—41no—
192C22×D3—81no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3—81no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C2×C22⋊C411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4C2×M1611yescharacteristic
205C2×C3×C4C2×M1611yescharacteristic
206C2×C3×C4#13111yescharacteristic
207C2×C3×C4#13111yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—41no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212D12—22no—
213D12—22no—
214D12—22no—
215D12—22no—
216D12—41no—
217D12—41no—
218D12—41no—
219D12—41no—
220C4×D3—44no—
221C4×D3—82no—
222C4×D3—82no—
223C4×D3—82no—
224C3×C8—22no—
225C3×C8—42no—
226C3×C8—41no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—61no—
231C24—61no—
232C2×D4—32no—
233C2×D4—32no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C22×C4—61no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—61no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—242no—
251C22×C4—242no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4C2×D1211yescharacteristic
254C22×C4#23511yescharacteristic
255D8—68no—
256M16—244no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—22no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—121no—
266C2×C8ΓL(1,𝔽25)12yescomplemented
267C4⋊C4—32no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C42#23911yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#38611yescharacteristic
282C22×C3#43511yescharacteristic
283C22×C3#43611yescharacteristic
284C2×D3—42no—
285C2×D3—42no—
286C2×D3—42no—
287C2×D3—42no—
288C2×D3—82no—
289C2×D3—82no—
290C2×D3—82no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—82no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4#40711yescharacteristic
299C3×C4#40811yescharacteristic
300C3×C4#43511yescharacteristic
301C3×C4#43611yescharacteristic
302Dic3—42no—
303Dic3—81no—
304C23—62no—
305C23—62no—
306C23—121no—
307C23—121no—
308C23—121no—
309C23—121no—
310C23—241no—
311C23—241no—
312C23—241no—
313C23—241no—
314C23C4×D1211yescharacteristic
315D4—62no—
316D4—62no—
317D4—62no—
318D4—62no—
319D4—121no—
320D4—121no—
321D4—121no—
322D4—121no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—62no—
326C2×C4—124no—
327C2×C4—121no—
328C2×C4—242no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4D3×C22⋊C411yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#89311yescharacteristic
337C2×C4#89411yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
340C2×C4#90111yescharacteristic
341C2×C4D3×M1611yescharacteristic
342C2×C4#90711yescharacteristic
343C8—22no—
344C8—42no—
345C8—41no—
346C8—81no—
347C8—122no—
348C8—241no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3#187011yescharacteristic
352C2×C3#190311yescharacteristic
353C2×C3#190411yescharacteristic
354D3—82no—
355D3—82no—
356C22—21no—
357C22—21no—
358C22—122no—
359C22—122no—
360C22—122no—
361C22—122no—
362C22—242no—
363C22—242no—
364C22—242no—
365C22—242no—
366C22—241no—
367C22—241no—
368C22#466511yescharacteristic
369C22#468811yescharacteristic
370C22#470611yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—122no—
376C4—241no—
377C4#472611yescharacteristic
378C4#472711yescharacteristic
379C4#483611yescharacteristic
380C4#483811yescharacteristic
381C3#909311yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2—242no—
385C2—242no—
386C2#6866911yescharacteristic
387C2#6902711yescharacteristic
388C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#386
prime radicalC22×D24C4
cosocleC2×C3×C42C23
hypercenter#1281D3
hypocenterC3#9093
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
1. #79288C2semidirect
3. #70783C2semidirect
4. #70757C2semidirect
5. #70756C2semidirect
6. C3×#1281C2semidirect
7. #67660C2semidirect
33. #1281D3semidirect
113. C2×D12M16semidirect
127. C2×C3×C8M16semidirect
266. C2×C8ΓL(1,𝔽25)semidirect
381. C3#9093semidirect

Maximal subgroups

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

subgroupindex
#792882
C2×#47112
#707832
#707572
#707562
C3×#12812
#676602
#90933