Finite Group Databasebeta

← #10691209

Subgroups — #10691209

Order 768.

524 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79048C211yesmaximal, characteristic
2#74193C211yesmaximal, characteristic
3#70782C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic, complemented
5#70506C211yesmaximal, characteristic, complemented
6C3×#1286C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9195—31nomaximal, Sylow
9C2×D3×M16—21no—
10C2×#897—21no—
11C2×#872C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5118—21no—
17#4973—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4839—22no—
21#4839—22no—
22#4674—21no—
23#4674C412yes—
24#4661—21no—
25#4661C412yes—
26#4486—22no—
27#4482—22no—
28#4469—21no—
29#4468—21no—
30#2905—31no—
31#2621—31no—
32#1546—31no—
33#1546—31no—
34#1458—31no—
35#1286—31no—
36#1286D311yescharacteristic, complemented
37C22×C4×D3D411yescharacteristic
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2311yescharacteristic
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4C2×C411yescharacteristic
44C2×(C2×D3)⋊C4D411yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3C2×C412yes—
48C2×Dic3⋊C4C2×C411yescharacteristic
49C2×Dic3⋊C4D411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51C2×#208—21no—
52D3×M16—42no—
53D3×M16—42no—
54C2×ΓL(1,𝔽25)—22no—
55C2×D3×C8—22no—
56#897—42no—
57#897—42no—
58#872—21no—
59#872—21no—
60#872—21no—
61#872—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3×#95—21no—
72C3×#95C2×C412yes—
73C3⋊#95—22no—
74C3⋊#95—22no—
75#830—21no—
76Dic3⋊C8—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C4×C3⋊C8—21no—
81C22×M16—61no—
82C2×C4⋊Q8—31no—
83C2×C22⋊Q8—61no—
84C2×#117—31no—
85#431—31no—
86#431C2×D311yescharacteristic
87#424—62no—
88#424—62no—
89#423—31no—
90#423—61no—
91#423C2×D311yescharacteristic
92#398—32no—
93#398—61no—
94#397—32no—
95#397—61no—
96#391—61no—
97#382C2×D311yescharacteristic
98#332—62no—
99#332—62no—
100#324—61no—
101#323—61no—
102C23×D3C22⋊C411yescharacteristic
103C22×C3×C4—22no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—22no—
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×C4×D3D812yes—
116C2×C4×D3SD1612yes—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×Dic6—41no—
122C2×Dic6—41no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—44no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—44no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147C4×Dic3—21no—
148C4×Dic3—21no—
149#208—41no—
150#208—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153ΓL(1,𝔽25)—44no—
154D3×C8—44no—
155C22×Q8—61no—
156C23×C4—31no—
157C2×M16—21no—
158C2×M16—62no—
159C2×M16—61no—
160C2×M16—122no—
161C2×M16—122no—
162C22×C8—62no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C22⋊Q8—122no—
168C22⋊Q8—122no—
169#117—61no—
170#117—61no—
171#117—61no—
172#117—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—61no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—62no—
182C2×C42—31no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—64no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8D1212yes—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—21no—
196#95—62no—
197#95—62no—
198#95C4×D312yes—
199C23×C3C2×C22⋊C411yescharacteristic
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3#10212yes—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4C2×D811yescharacteristic
217C2×C3×C4C2×SD1611yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3#10212yes—
230C4×D3—22no—
231C4×D3—22no—
232C4×D3—22no—
233C4×D3—22no—
234Dic6—81no—
235Dic6—81no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C24—31no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—121no—
246C2×Q8—121no—
247C2×Q8—121no—
248C2×Q8—121no—
249C22×C4—22no—
250C22×C4—31no—
251C22×C4—32no—
252C22×C4—32no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—62no—
260C22×C4C2×C4×D311yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263M16—41no—
264M16—41no—
265M16—124no—
266M16—121no—
267M16—121no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—62no—
271C2×C8—124no—
272C2×C8—121no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—62no—
280C4⋊C4—62no—
281C4⋊C4—124no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—124no—
289C42—61no—
290C42—61no—
291C42C2×D1211yescharacteristic
292C42#23511yescharacteristic
293C42D3×D411yescharacteristic
294C42#23911yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3#38611yescharacteristic
298C22×C3C2×#10211yescharacteristic
299C22×C3#41711yescharacteristic
300C2×D3—22no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—42no—
304C2×D3—81no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—82no—
310C3×C4#44711yescharacteristic
311C3×C4#44811yescharacteristic
312C3×C4#48011yescharacteristic
313C3×C4#48111yescharacteristic
314Dic3—22no—
315Dic3—41no—
316Dic3—41no—
317Dic3—82no—
318C23—32no—
319C23—61no—
320C23—61no—
321C23—121no—
322C23—121no—
323C23D3×C22⋊C411yescharacteristic
324Q8—241no—
325Q8—241no—
326Q8—241no—
327Q8—241no—
328C2×C4—32no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4C4×D1211yescharacteristic
348C2×C4#88211yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#89311yescharacteristic
351C2×C4#89411yescharacteristic
352C2×C4#89511yescharacteristic
353C2×C4D3×D811yescharacteristic
354C2×C4D3×SD1611yescharacteristic
355C2×C4#91411yescharacteristic
356C2×C4#91811yescharacteristic
357C8—42no—
358C8—81no—
359C8—122no—
360C8—241no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3#187411yescharacteristic
364C2×C3#190911yescharacteristic
365C2×C3#191511yescharacteristic
366D3—42no—
367C22—21no—
368C22—21no—
369C22—62no—
370C22—122no—
371C22—121no—
372C22—121no—
373C22—241no—
374C22#466511yescharacteristic
375C22D3×#10211yescharacteristic
376C22#480311yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—62no—
382C4—82no—
383C4—121no—
384C4—121no—
385C4—242no—
386C4#477311yescharacteristic
387C4#480611yescharacteristic
388C4#486311yescharacteristic
389C4#488211yescharacteristic
390C3#919511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—122no—
394C2#6871211yescharacteristic
395C2#6888011yescharacteristic
396C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9195
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
4. #70778C2semidirect
5. #70506C2semidirect
6. C3×#1286C2semidirect
7. #67631C2semidirect
36. #1286D3semidirect
390. C3#9195semidirect

Maximal subgroups

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

subgroupindex
#790482
#741932
#707822
#707782
#705062
C3×#12862
#676312
#91953