Finite Group Databasebeta

← #10691207

Subgroups — #10691207

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#83261C211yesmaximal, characteristic
2#79048C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic, complemented
4#71773C211yesmaximal, characteristic, complemented
5#70782C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9195—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—21no—
11C2×D3×M16—21no—
12C2×#872C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5229—21no—
16#5118—21no—
17#4992—21no—
18#4992C412yes—
19#4983—21no—
20#4983C412yes—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4839—22no—
25#4839—22no—
26C3×#332—22no—
27#4547—22no—
28#4469—21no—
29#4468—21no—
30#2905—31no—
31#2621—31no—
32#1546—31no—
33#1546D311yescharacteristic, complemented
34#1458—31no—
35#1286—31no—
36#1286—31no—
37C22×C3×Q8—21no—
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C412yes—
46C2×C3×C42C2311yescharacteristic
47#946—42no—
48#946—42no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4C2×C411yescharacteristic
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—22no—
54C2×Dic3⋊C4C2×C411yescharacteristic
55C2×Dic3⋊C4D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×#208—21no—
58D3×M16—42no—
59D3×M16—42no—
60C2×ΓL(1,𝔽25)—22no—
61C2×D3×C8—22no—
62#872—21no—
63#872—21no—
64#872—21no—
65#872—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3⋊#95—21no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95C2×C412yes—
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⋊Q8C2×D311yescharacteristic
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#382—31no—
98#332—22no—
99#332—62no—
100#324—61no—
101#323—61no—
102C23×D3C22⋊C411yescharacteristic
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—22no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—22no—
114C22×Dic3C22⋊C411yescharacteristic
115C22×Dic3C22⋊C411yescharacteristic
116C22×Dic3C22⋊C411yescharacteristic
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3D812yes—
122C2×C4×D3SD1612yes—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—44no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142Dic3⋊C4—21no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146Dic3⋊C4—44no—
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—21no—
156C23×C4—31no—
157C2×M16—21no—
158C2×M16—62no—
159C2×M16—61no—
160C2×M16—122no—
161C2×M16—122no—
162C22×C8—62no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
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—62no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4C4×D312yes—
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⋊C8#6712yes—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—32no—
196#95—62no—
197#95—62no—
198#95—61no—
199C23×C3C2×C22⋊C411yescharacteristic
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3#10212yes—
205C3×Q8—81no—
206C3×Q8—81no—
207C3×Q8—81no—
208C3×Q8—81no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4C2×D811yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3#10212yes—
234C4×D3—22no—
235C4×D3—22no—
236C4×D3—22no—
237C4×D3—22no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C24—31no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
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—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
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—
291C42#23511yescharacteristic
292C42D3×D411yescharacteristic
293C42#23711yescharacteristic
294C42C2×#6711yescharacteristic
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—81no—
325Q8—81no—
326Q8—81no—
327Q8—81no—
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×C4#67⋊C411yescharacteristic
348C2×C4#88311yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4D3×D811yescharacteristic
351C2×C4#91111yescharacteristic
352C2×C4D3×SD1611yescharacteristic
353C2×C4#91511yescharacteristic
354C2×C4C4×#6711yescharacteristic
355C2×C4#93711yescharacteristic
356C2×C4#93811yescharacteristic
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—
374C22D3×#10211yescharacteristic
375C22#477111yescharacteristic
376C22#495611yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—62no—
382C4—82no—
383C4—121no—
384C4—121no—
385C4—242no—
386C4#477411yescharacteristic
387C4#477711yescharacteristic
388C4#515611yescharacteristic
389C4#517011yescharacteristic
390C3#919511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—122no—
394C2#7012411yescharacteristic
395C2#7020011yescharacteristic
396C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9195
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
3. C3×#1546C2semidirect
4. #71773C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
33. #1546D3semidirect
390. C3#9195semidirect

Maximal subgroups

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

subgroupindex
#832612
#790482
C3×#15462
#717732
#707822
#677382
#677132
#91953