Finite Group Databasebeta

← #10693300

Subgroups — #10693300

Order 768.

444 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82311C211yesmaximal, characteristic
2#79326C211yesmaximal, characteristic, complemented
3C3×#1466C211yesmaximal, characteristic, complemented
4#71599C211yesmaximal, characteristic, complemented
5#70809C211yesmaximal, characteristic, complemented
6C3⋊#1308C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#9532—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5479C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5218C411yescharacteristic
14#5217C411yescharacteristic
15#5018—21no—
16#5016—21no—
17#5014—21no—
18#5009—21no—
19#4971C411yescharacteristic
20#4967C411yescharacteristic
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4836—21no—
25#4828—21no—
26#4729—21no—
27#4728—21no—
28#4709—21no—
29#4708—21no—
30C3×#323—21no—
31C3×#323—21no—
32D4⋊(C3⋊C8)—21no—
33D4⋊(C3⋊C8)—21no—
34#4465—21no—
35#4464—21no—
36#3114—31no—
37#2967—31no—
38#1535—31no—
39#1490—31no—
40#1466D311yescharacteristic, complemented
41#1308—31no—
42#1291—31no—
43C22×C3×D4C2×C411yescharacteristic
44C23×C3×C4—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#214—21no—
53C2×#213—21no—
54#930C2×C411yescharacteristic
55#929—21no—
56C4×#67—21no—
57#924C2×C411yescharacteristic
58C2×#208—21no—
59C2×ΓL(1,𝔽25)—21no—
60C2×D3×C8—21no—
61#876—21no—
62#875D411yescharacteristic
63C4×D12D411yescharacteristic
64#872—21no—
65#870D411yescharacteristic
66C4×Dic6D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D411yescharacteristic
71C3×C22⋊C8D411yescharacteristic
72C3×#97—21no—
73C3×#97—21no—
74#832—21no—
75#832—21no—
76#830—21no—
77#830C2×C411yescharacteristic
78#830C2×C411yescharacteristic
79(C2×D3)⋊C8—21no—
80(C2×D3)⋊C8—21no—
81#815—21no—
82Dic3⋊C8—21no—
83Dic3⋊C8—21no—
84C8×Dic3—21no—
85#810—21no—
86#809—21no—
87#808—21no—
88#807—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91#803—21no—
92#803—21no—
93#803—21no—
94C4×C3⋊C8D411yescharacteristic
95C4×C3⋊C8D411yescharacteristic
96#529—31no—
97C2×C4×D4C2×D311yescharacteristic
98#483—31no—
99#480—31no—
100#478—61no—
101#463—61no—
102#450—31no—
103#447—31no—
104#446—61no—
105#442—61no—
106#440—61no—
107#436—61no—
108#435—61no—
109#435—61no—
110#433—61no—
111#432—31no—
112#432—31no—
113#432C2×D311yescharacteristic
114#432C2×D311yescharacteristic
115#405—61no—
116C4×M16—31no—
117#330—61no—
118#329—61no—
119D4⋊C8—61no—
120D4⋊C8—61no—
121#323—21no—
122#323—21no—
123C24×C3—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—82no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C2×#67—21no—
135C2×D12C22⋊C411yescharacteristic
136C2×C4×D3—21no—
137C2×Dic6C22⋊C411yescharacteristic
138C2×C3×C8—21no—
139C2×C3×C8—21no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—21no—
150#214—42no—
151#213—42no—
152(C2×D3)⋊C4—21no—
153(C2×D3)⋊C4—21no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156C4⋊Dic3C22⋊C411yescharacteristic
157C4⋊Dic3C22⋊C411yescharacteristic
158Dic3⋊C4—21no—
159Dic3⋊C4—21no—
160Dic3⋊C4—21no—
161Dic3⋊C4—21no—
162C4×Dic3—21no—
163#208—42no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—41no—
169ΓL(1,𝔽25)—42no—
170D3×C8—42no—
171C22×D4C4×D311yescharacteristic
172C23×C4—21no—
173C2×SD16—61no—
174C2×D8—61no—
175C2×M16—61no—
176C2×M16—61no—
177C22×C8—61no—
178#126—61no—
179#125—31no—
180#124—31no—
181#123—61no—
182C22⋊Q8—31no—
183#121—31no—
184C4×Q8—31no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—31no—
190C4×D4—61no—
191#117—61no—
192C2×C4⋊C4C4×D311yescharacteristic
193C2×C22⋊C4—21no—
194C2×C42C22×D311yescharacteristic
195#107—61no—
196#106—61no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203#103—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—31no—
209C22⋊C8—31no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8#6711yescharacteristic
214C22⋊C8#6711yescharacteristic
215#97—21no—
216#97—21no—
217#97—61no—
218#97—61no—
219#97—61no—
220#97—61no—
221C4×C8—31no—
222C4×C8—31no—
223C4×C8—61no—
224C23×C3—22no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3C4×D411yescharacteristic
230C22×D3—21no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—82no—
240C2×C3×C4—82no—
241C2×C3×C4—81no—
242C2×C3×C4C2×C22⋊C411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248#67—81no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252D12—41no—
253C4×D3—41no—
254C4×D3—41no—
255Dic6—41no—
256C3×C8—41no—
257C3×C8—41no—
258C3×C8—81no—
259C3⋊C8—22no—
260C3⋊C8—41no—
261C3⋊C8—41no—
262C3⋊C8—41no—
263C3⋊C8—81no—
264C24—21no—
265C2×Q8—31no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—31no—
269C2×D4—42no—
270C2×D4—61no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—61no—
275C22×C4—82no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4#23511yescharacteristic
278C22×C4C2×#6711yescharacteristic
279SD16—122no—
280D8—122no—
281M16—122no—
282M16—122no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—122no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C42—61no—
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C42#23711yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—42no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#38611yescharacteristic
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—81no—
331Dic3—41no—
332Dic3—41no—
333Dic3—41no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—61no—
338C23—81no—
339C23—81no—
340C23C4×#6711yescharacteristic
341Q8—121no—
342D4—22no—
343D4—22no—
344D4—41no—
345D4—41no—
346D4—121no—
347D4—241no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4D3×C22⋊C411yescharacteristic
361C2×C4#67⋊C411yescharacteristic
362C2×C4#88311yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4#93711yescharacteristic
365C2×C4#93811yescharacteristic
366C8—41no—
367C8—41no—
368C8—62no—
369C8—81no—
370C8—121no—
371C8—121no—
372C8—121no—
373C8—241no—
374C2×C3—41no—
375C2×C3—82no—
376C2×C3#189111yescharacteristic
377C2×C3#190311yescharacteristic
378C2×C3#190611yescharacteristic
379D3—81no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22—81no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22#495611yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—81no—
399C4—121no—
400C4—121no—
401C4—121no—
402C3#953211yescharacteristic, complemented, Sylow
403C2—41no—
404C2—82no—
405C2—241no—
406C2#6999211yescharacteristic
407C2#7005511yescharacteristic
408C2#7008711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1466C2
socleC22×C3#386
prime radical#5217C4
cosocleC2×C3×C42C23
hypercenter#1466D3
hypocenterC3#9532
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #79326C2semidirect
3. C3×#1466C2semidirect
4. #71599C2semidirect
5. #70809C2semidirect
6. C3⋊#1308C2semidirect
7. #67753C2semidirect
40. #1466D3semidirect
402. C3#9532semidirect

Maximal subgroups

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

subgroupindex
#823112
#793262
C3×#14662
#715992
#708092
C3⋊#13082
#677532
#95323