Finite Group Databasebeta

← #10690061

Subgroups — #10690061

Order 768.

476 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79049C211yesmaximal, characteristic
2#79025C211yesmaximal, characteristic
3#70768C211yesmaximal, characteristic, complemented
4#70754C211yesmaximal, characteristic
5#70501C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8993—31nomaximal, Sylow
9C2×#907—21no—
10C4×#235C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5119—21no—
16#5113—21no—
17#5106—21no—
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20#4851C411yescharacteristic
21#4850C411yescharacteristic
22#4847—21no—
23#4846C411yescharacteristic
24#4842C411yescharacteristic
25#4839—22no—
26#4837—21no—
27D3×C4⋊C8—21no—
28#4829—21no—
29#4484—21no—
30#4484—22no—
31#4483—21no—
32#4456—21no—
33#4456—21no—
34#3076—31no—
35#2906—31no—
36#1493—31no—
37#1481—31no—
38#1476—31no—
39#1301D311yescharacteristic, complemented
40#1277—31no—
41C2×#235D411yescharacteristic
42C2×C3×M16—21no—
43C22×C3×C8—21no—
44C2×C3×C42C2311yescharacteristic
45C4×#67—21no—
46C4×#67—21no—
47#926—21no—
48#926D411yescharacteristic
49#924—21no—
50#924—21no—
51C22×C3⋊C8—21no—
52C22×C3⋊C8—21no—
53#907—42no—
54#907—42no—
55C2×ΓL(1,𝔽25)—22no—
56C2×D3×C8—22no—
57C4×D12C2×C411yescharacteristic
58C4×D12D411yescharacteristic
59#872—21no—
60C42×D3—21no—
61#870—21no—
62#869C2×C411yescharacteristic
63#869C2×C411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6C2×C411yescharacteristic
67C4×Dic6D411yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8C2×C411yescharacteristic
72C3×C4⋊C8C2×C411yescharacteristic
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76#821—21no—
77#821—21no—
78#818—21no—
79#818—21no—
80#817—21no—
81#817—21no—
82#816—21no—
83#816—21no—
84Dic3⋊C8—22no—
85Dic3⋊C8—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D411yescharacteristic
88C4⋊(C3⋊C8)D411yescharacteristic
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×#131—61no—
92#533—31no—
93C4×#41—31no—
94#479—61no—
95#478—61no—
96#477—31no—
97#476—31no—
98#475—31no—
99#474—31no—
100#431C2×D311yescharacteristic
101#427—61no—
102#424—61no—
103#424—62no—
104#423—61no—
105#423C2×D311yescharacteristic
106C2×C4⋊C8—31no—
107C2×C4⋊C8—61no—
108C2×C4⋊C8C2×D311yescharacteristic
109#416—61no—
110C2×C4×C8—31no—
111Q8⋊C8—61no—
112Q8⋊C8—61no—
113Q8⋊C8—61no—
114Q8⋊C8—62no—
115D4⋊C8—61no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—21no—
120#235—21no—
121#235—21no—
122#235SD1612yes—
123#235Q1612yes—
124C2×D12C22⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6C22⋊C411yescharacteristic
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C42C22×C411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#217—21no—
144#217—41no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3C22⋊C411yescharacteristic
152C4⋊Dic3C22⋊C411yescharacteristic
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168ΓL(1,𝔽25)—44no—
169D3×C8—44no—
170C2×#41—31no—
171#131—122no—
172#131—122no—
173C2×M16—21no—
174C2×M16—62no—
175C22×C8—21no—
176C22×C8—62no—
177C22×C8—61no—
178C22×C8—61no—
179C4⋊Q8—31no—
180C4⋊Q8—31no—
181#125—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C4×Q8—31no—
185C4×Q8—31no—
186C4×Q8—61no—
187C4×Q8—61no—
188C4×D4—31no—
189C4×D4—31no—
190C4×D4—61no—
191C4×D4—61no—
192#117—31no—
193#117—61no—
194#117—61no—
195C2×C42—61no—
196C2×C42C22×D311yescharacteristic
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—31no—
205C4⋊C8—31no—
206C4⋊C8—62no—
207C4⋊C8—62no—
208C4⋊C8—61no—
209C4⋊C8C4×D311yescharacteristic
210C4⋊C8C4×D311yescharacteristic
211C4⋊C8D1212yes—
212#103—61no—
213#103—61no—
214#102—61no—
215#102—61no—
216C22⋊C8—21no—
217C4×C8—21no—
218C4×C8—61no—
219C4×C8—61no—
220C23×C3#12311yescharacteristic
221C22×D3—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4C2×SD1611yescharacteristic
232C2×C3×C4C2×Q1611yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4#13711yescharacteristic
235#67—22no—
236#67—22no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242D12—21no—
243D12#10312yes—
244C4×D3—22no—
245C4×D3—22no—
246C4×D3—41no—
247C4×D3—41no—
248C4×D3—81no—
249Dic6—21no—
250Dic6—42no—
251Dic6—41no—
252Dic6—41no—
253Dic6#10312yes—
254C3×C8—41no—
255C3×C8—41no—
256C3×C8—42no—
257C3×C8—81no—
258C3⋊C8—42no—
259C3⋊C8—41no—
260C3⋊C8—41no—
261#41—32no—
262#41—32no—
263#41—61no—
264#41—61no—
265C2×Q8—31no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—31no—
269C2×D4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4#23511yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4#23911yescharacteristic
276M16—41no—
277M16—41no—
278M16—124no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—124no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C4⋊C4—31no—
294C4⋊C4—31no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—121no—
307C42—61no—
308C42—61no—
309C42—61no—
310C42—121no—
311C42C2×C4×D311yescharacteristic
312C42C2×D1211yescharacteristic
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3#48311yescharacteristic
319C22×C3#48411yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4C2×#10311yescharacteristic
327C3×C4#41711yescharacteristic
328C3×C4#44311yescharacteristic
329C3×C4#44411yescharacteristic
330C3×C4#45011yescharacteristic
331C3×C4#45111yescharacteristic
332Dic3—22no—
333Dic3—41no—
334Dic3—41no—
335Dic3—81no—
336Dic3—81no—
337C23—61no—
338C23#89411yescharacteristic
339Q8—32no—
340Q8—61no—
341Q8—122no—
342Q8—121no—
343Q8—121no—
344D4—32no—
345D4—62no—
346D4—62no—
347D4—61no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—61no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—241no—
361C2×C4C4×D1211yescharacteristic
362C2×C4D3×C22⋊C411yescharacteristic
363C2×C4#88211yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4#89311yescharacteristic
366C2×C4#89511yescharacteristic
367C2×C4#91111yescharacteristic
368C2×C4D3×SD1611yescharacteristic
369C2×C4#91511yescharacteristic
370C2×C4D3×Q1611yescharacteristic
371C8—41no—
372C8—41no—
373C8—42no—
374C8—81no—
375C8—122no—
376C8—121no—
377C8—121no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3#187211yescharacteristic
381C2×C3#187611yescharacteristic
382C2×C3#188111yescharacteristic
383D3—42no—
384C22—21no—
385C22—21no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22#466511yescharacteristic
390C22#486211yescharacteristic
391C22#488411yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—62no—
396C4—121no—
397C4—121no—
398C4—241no—
399C4—241no—
400C4#477111yescharacteristic
401C4#477611yescharacteristic
402C4D3×#10311yescharacteristic
403C4#480911yescharacteristic
404C4#489411yescharacteristic
405C4#489511yescharacteristic
406C3#899311yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—122no—
410C2#6887811yescharacteristic
411C2#6894611yescharacteristic
412C2#6903511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#8993
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70768C2semidirect
5. #70501C2semidirect
6. C3×#1301C2semidirect
7. #67703C2semidirect
39. #1301D3semidirect
406. C3#8993semidirect

Maximal subgroups

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

subgroupindex
#790492
#790252
#707682
#707542
#705012
C3×#13012
#677032
#89933