Finite Group Databasebeta

← #10691051

Subgroups — #10691051

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79060C211yesmaximal, characteristic
2#79012C211yesmaximal, characteristic
3#70780C211yesmaximal, characteristic, complemented
4#70755C211yesmaximal, characteristic
5C2×#4481C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9169—31nomaximal, Sylow
9C2×#901—21no—
10C4×#235C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5133—21no—
16#5121—21no—
17#5102—21no—
18#4924C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4851C412yes—
21#4850—21no—
22#4846C412yes—
23#4842—21no—
24#4839—22no—
25#4837—21no—
26D3×C4⋊C8—21no—
27#4482—22no—
28#4481—21no—
29#4481—21no—
30#4459—21no—
31#4456—21no—
32#3079—31no—
33#2906—31no—
34#1549—31no—
35C2×#333—31no—
36#1477—31no—
37#1301D311yescharacteristic, complemented
38#1292—31no—
39C2×#235D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44C4×#67—21no—
45C4×#67—21no—
46#926D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51#901—42no—
52#901—42no—
53C2×ΓL(1,𝔽25)—22no—
54C2×D3×C8—22no—
55C4×D12C2×C411yescharacteristic
56C4×D12D411yescharacteristic
57#872—21no—
58C42×D3—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869C2×C412yes—
63C4×Dic6C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×C4⋊C8D411yescharacteristic
70C3×C4⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#821—22no—
74#818—22no—
75#817—22no—
76#816—22no—
77Dic3⋊C8—22no—
78Dic3⋊C8—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C2×#131—61no—
84C2×C4⋊Q8—31no—
85C4×#41—31no—
86#477—61no—
87#476—61no—
88#475—32no—
89#474—32no—
90#431C2×D311yescharacteristic
91#428—61no—
92#424—62no—
93#424—61no—
94#423—31no—
95#423—61no—
96#423C2×D311yescharacteristic
97C2×C4⋊C8—31no—
98C2×C4⋊C8—61no—
99C2×C4⋊C8C2×D311yescharacteristic
100#417—61no—
101#333—61no—
102#333—61no—
103#332—62no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—21no—
110C22×Dic3—41no—
111#235—21no—
112#235—21no—
113#235SD1612yes—
114#235Q1612yes—
115C2×D12C22⋊C411yescharacteristic
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×Dic6—22no—
120C2×Dic6—41no—
121C2×Dic6—82no—
122C2×Dic6C22⋊C411yescharacteristic
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C3×C42C22×C411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135#217—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3C22⋊C411yescharacteristic
143C4⋊Dic3C22⋊C411yescharacteristic
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146C4×Dic3—21no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149#208—41no—
150#208—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—81no—
154ΓL(1,𝔽25)—44no—
155D3×C8—44no—
156C2×#41—31no—
157C22×Q8—61no—
158#131—122no—
159#131—122no—
160C2×M16—21no—
161C2×M16—62no—
162C2×M16—61no—
163C22×C8—21no—
164C22×C8—62no—
165C22×C8—61no—
166C4⋊Q8—32no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4×Q8—31no—
171C4×Q8—31no—
172C4×D4—31no—
173C4×D4—31no—
174C4×D4—61no—
175C4×D4—61no—
176#117—31no—
177#117—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—62no—
189C4⋊C8—62no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8C4×D312yes—
194C4⋊C8D1211yescharacteristic
195C4⋊C8D1211yescharacteristic
196#103—62no—
197#102—62no—
198C22⋊C8—21no—
199C4×C8—21no—
200C23×C3#12311yescharacteristic
201C22×D3—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4C2×Q1611yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4#13711yescharacteristic
215#67—22no—
216#67—22no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—42no—
221C2×Dic3—82no—
222D12—21no—
223D12#10312yes—
224C4×D3—22no—
225C4×D3—22no—
226C4×D3—41no—
227C4×D3—41no—
228C4×D3—81no—
229Dic6—21no—
230Dic6—82no—
231Dic6—82no—
232Dic6#10312yes—
233C3×C8—42no—
234C3×C8—41no—
235C3×C8—41no—
236C3×C8—81no—
237C3⋊C8—42no—
238C3⋊C8—81no—
239#41—32no—
240#41—32no—
241#41—61no—
242#41—61no—
243C2×Q8—31no—
244C2×Q8—62no—
245C2×Q8—121no—
246C2×Q8—242no—
247C2×D4—31no—
248C2×D4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4#23511yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4#23911yescharacteristic
256M16—41no—
257M16—41no—
258M16—124no—
259M16—121no—
260M16—121no—
261C2×C8—22no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—62no—
268C2×C8—124no—
269C2×C8—121no—
270C2×C8—241no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42C2×C4×D311yescharacteristic
286C42C2×D1211yescharacteristic
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C22×C3—21no—
290C22×C3—21no—
291C22×C3#38611yescharacteristic
292C22×C3#48311yescharacteristic
293C22×C3#48411yescharacteristic
294C2×D3—22no—
295C2×D3—41no—
296C2×D3—41no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4C2×#10311yescharacteristic
301C3×C4#41711yescharacteristic
302C3×C4#45011yescharacteristic
303C3×C4#45111yescharacteristic
304C3×C4#47111yescharacteristic
305C3×C4#47311yescharacteristic
306Dic3—22no—
307Dic3—41no—
308Dic3—41no—
309Dic3—82no—
310C23—61no—
311C23#89411yescharacteristic
312Q8—32no—
313Q8—61no—
314Q8—242no—
315Q8—242no—
316D4—32no—
317D4—62no—
318D4—62no—
319D4—61no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—241no—
332C2×C4—242no—
333C2×C4C4×D1211yescharacteristic
334C2×C4D3×C22⋊C411yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#89311yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4#91111yescharacteristic
340C2×C4D3×SD1611yescharacteristic
341C2×C4#91511yescharacteristic
342C2×C4D3×Q1611yescharacteristic
343C8—42no—
344C8—41no—
345C8—41no—
346C8—81no—
347C8—122no—
348C8—241no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3#187211yescharacteristic
352C2×C3#190311yescharacteristic
353C2×C3#190811yescharacteristic
354D3—42no—
355C22—21no—
356C22—21no—
357C22—62no—
358C22—121no—
359C22—121no—
360C22#466511yescharacteristic
361C22#486211yescharacteristic
362C22#488411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—62no—
367C4—121no—
368C4—121no—
369C4—242no—
370C4#477111yescharacteristic
371C4#477611yescharacteristic
372C4D3×#10311yescharacteristic
373C4#480911yescharacteristic
374C4#489611yescharacteristic
375C4#489811yescharacteristic
376C3#916911yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—122no—
380C2#6887811yescharacteristic
381C2#6894911yescharacteristic
382C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9169
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70780C2semidirect
5. C2×#4481C2semidirect
6. C3×#1301C2semidirect
7. #67711C2semidirect
37. #1301D3semidirect
376. C3#9169semidirect

Maximal subgroups

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

subgroupindex
#790602
#790122
#707802
#707552
C2×#44812
C3×#13012
#677112
#91693