Finite Group Databasebeta

← #10691464

Subgroups — #10691464

Order 768.

550 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82319C211yesmaximal, characteristic
2#82314C211yesmaximal, characteristic
3C3×#1486C211yesmaximal, characteristic
4#70792C211yesmaximal, characteristic, complemented
5#70514C211yesmaximal, characteristic, complemented
6#67736C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9235—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#926—21no—
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5221—21no—
15#5218—21no—
16C4×#217—21no—
17#5019C412yescomplemented
18C4×#214—22no—
19#5008C412yescomplemented
20#4984—21no—
21#4971—21no—
22#4951—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26C3×D4⋊C8—22no—
27#4555—21no—
28#4555—21no—
29#4467—22no—
30#3101—31no—
31#2293—31no—
32#1994—31no—
33#1503—31no—
34#1486D311yescharacteristic
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×C3×M16—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4C2×C412yes—
43C3×C4×D4D412yes—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#217—22no—
48C2×#217—22no—
49C2×#214—22no—
50#926—21no—
51#926—42no—
52#926—42no—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3—21no—
55#924—21no—
56#924—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3D411yescharacteristic
61C2×C4×Dic3D411yescharacteristic
62#870—21no—
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#832—22no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#810—22no—
77#808—22no—
78C4⋊(C3⋊C8)C2×C412yes—
79C4×C3⋊C8—21no—
80#533—31no—
81C2×C4×D4C2×D311yescharacteristic
82C2×#117—61no—
83#483—61no—
84#476—32no—
85#461—32no—
86#450—61no—
87C4×SD16—62no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423C2×D311yescharacteristic
91#390—61no—
92#389—61no—
93#388—62no—
94C4×C4⋊C4—31no—
95C4×C22⋊C4—61no—
96C43—31no—
97#339—61no—
98#339—61no—
99#328—62no—
100D4⋊C8—22no—
101C24×C3—21no—
102C2×C3×D4—21no—
103C2×C3×D4—42no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4C22⋊C412yes—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132#217—44no—
133#217—44no—
134#217—41no—
135#214—44no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—44no—
141C4⋊Dic3—42no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C4×Dic3—21no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147C4×Dic3—42no—
148C4×Dic3—42no—
149C4×Dic3—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3SD1612yescomplemented
153C2×C3⋊C8—22no—
154C22×D4D1211yescharacteristic
155C23×C4—21no—
156C2×SD16—62no—
157C2×M16—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4C4×D312yes—
168C4×D4#6712yes—
169#117—61no—
170#117—122no—
171#117—122no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4D1211yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C42—31no—
179C2×C42—31no—
180C2×C42—61no—
181C2×C42—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189#103—62no—
190#102—62no—
191C22⋊C8—21no—
192C22⋊C8—61no—
193C4×C8—21no—
194C4×C8—61no—
195#95—32no—
196#95—62no—
197C23×C3—22no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3#12011yescharacteristic
203C3×D4—22no—
204C3×D4—22no—
205C3×D4—42no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236Dic6—42no—
237Dic6—81no—
238C3×C8—81no—
239C3×C8—81no—
240C3⋊C8—42no—
241C24—21no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—21no—
245C2×D4—42no—
246C2×D4(C2×D3)⋊C412yes—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—62no—
257C22×C4C2×D1211yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260SD16—124no—
261M16—41no—
262M16—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—62no—
266C4⋊C4—21no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—122no—
272C4⋊C4—124no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4(C2×D3)⋊C412yes—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—124no—
279C22⋊C4—124no—
280C22⋊C4—121no—
281C42—32no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—122no—
286C42—122no—
287C42—121no—
288C42—121no—
289C42—121no—
290C42C2×C4×D311yescharacteristic
291C42#23711yescharacteristic
292C42#23711yescharacteristic
293C42C2×#6711yescharacteristic
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—82no—
300C22×C3—82no—
301C22×C3#38611yescharacteristic
302C22×C3#45011yescharacteristic
303C22×C3#45211yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4#42112yes—
309C3×C4C4×SD1611yescharacteristic
310C3×C4#44011yescharacteristic
311C3×C4#46111yescharacteristic
312C3×C4#46311yescharacteristic
313Dic3—42no—
314Dic3—42no—
315Dic3—82no—
316Dic3—81no—
317Dic3—81no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23#88211yescharacteristic
324Q8—122no—
325Q8—241no—
326D4—22no—
327D4—22no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4#67⋊C411yescharacteristic
350C2×C4#67⋊C411yescharacteristic
351C2×C4#88611yescharacteristic
352C2×C4#91111yescharacteristic
353C2×C4#91211yescharacteristic
354C2×C4D3×SD1611yescharacteristic
355C2×C4#91511yescharacteristic
356C2×C4C2×(C2×D3)⋊C411yescharacteristic
357C2×C4#93911yescharacteristic
358C2×C4#93911yescharacteristic
359C8—81no—
360C8—81no—
361C8—122no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—82no—
365C2×C3#187711yescharacteristic
366C2×C3#189711yescharacteristic
367C2×C3#190611yescharacteristic
368C22—21no—
369C22—21no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—82no—
374C22—82no—
375C22#477611yescharacteristic
376C22#478311yescharacteristic
377C22#495811yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—122no—
383C4—122no—
384C4—242no—
385C4—241no—
386C4—241no—
387C4#475711yescharacteristic
388C4#475811yescharacteristic
389C4#513912yes—
390C4#515311yescharacteristic
391C4#516211yescharacteristic
392C3#923511yescharacteristic, complemented, Sylow
393C2—21no—
394C2—21no—
395C2—82no—
396C2#6996611yescharacteristic
397C2#7017111yescharacteristic
398C2#7039511yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9235
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
4. #70792C2semidirect
5. #70514C2semidirect
6. #67736C2semidirect
7. #67715C2semidirect
17. #5019C4semidirect
19. #5008C4semidirect
152. C4×Dic3SD16semidirect
392. C3#9235semidirect

Maximal subgroups

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

subgroupindex
#823192
#823142
C3×#14862
#707922
#705142
#677362
#677152
#92353