Finite Group Databasebeta

← #11713124

Subgroups — #11713124

Order 768.

502 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×C4⋊#226C211yesmaximal, characteristic
2C2×#5388C211yesmaximal, characteristic
3C2×C4×#226C211yesmaximal, characteristic
4#2351⋊C3C211yesmaximal, characteristic
5#73063C211yesmaximal, characteristic
6#73061C211yesmaximal, characteristic
7#73061C211yesmaximal, characteristic
8#23014—31nomaximal, Sylow
9C22×#226C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C4⋊#226C414yescomplemented
17#5388—22no—
18C4×#226—22no—
19#4654—41nomaximal
20C2×#462—31no—
21C2×#441—31no—
22C2×C4×Q16—31no—
23#2365—31no—
24#2351D311yescharacteristic
25#1960—31no—
26#1949—31no—
27#1935—31no—
28#1933—31no—
29#1878—31no—
30#1878—31no—
31#1865—31no—
32#1863—31no—
33#1771—31no—
34#1763—31no—
35C22×SL(2,𝔽3)C2311yescharacteristic
36C2×#226—21no—
37C2×#226—21no—
38C2×#226—21no—
39C2×#226—21no—
40C2×#226C2×C412yes—
41C2×#226C2×C412yes—
42C2×C4⋊Dic3—41no—
43C2×Dic3⋊C4—41no—
44C2×C4×Dic3—41no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)—21no—
47C4×SL(2,𝔽3)C2×C412yes—
48#859—21no—
49#859D412yes—
50C3×#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C3⋊#95—41no—
54C22×Q16—31no—
55C22×Q16—31no—
56C2×C4⋊Q8—31no—
57C2×C4×Q8—31no—
58C2×C4×Q8—31no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×Q8C2×D311yescharacteristic
61C2×C4×Q8C2×D311yescharacteristic
62#462—34no—
63#462—34no—
64#441—62no—
65#441—62no—
66C4×Q16—62no—
67C4×Q16—62no—
68C2×#107—31no—
69C2×#106—31no—
70C2×C4⋊C8—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#103—31no—
78C2×#103—31no—
79C2×#97—31no—
80C2×#97—31no—
81C2×C4×C8—31no—
82#391—31no—
83#391—31no—
84#389—31no—
85#389—31no—
86#389—61no—
87#387—61no—
88#387—61no—
89#382—31no—
90C4×C4⋊C4—31no—
91C4×C4⋊C4—31no—
92C4×C4⋊C4—61no—
93#340—31no—
94#340—31no—
95#340—31no—
96#340—31no—
97#336—31no—
98#336—31no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C2×SL(2,𝔽3)C22×C411yescharacteristic
107C2×SL(2,𝔽3)C2×D411yescharacteristic
108C2×SL(2,𝔽3)#4111yescharacteristic
109#226—22no—
110#226—22no—
111C4⋊Dic3—44no—
112Dic3⋊C4—82no—
113C4×Dic3—82no—
114C22×Q8—31no—
115C22×Q8—31no—
116C22×Q8C22×D311yescharacteristic
117C2×Q16—32no—
118C2×Q16—32no—
119C2×Q16—32no—
120C2×Q16—32no—
121C2×Q16—61no—
122C2×Q16—61no—
123C2×Q16—61no—
124C2×Q16—61no—
125C2×Q16—61no—
126C2×Q16—61no—
127C2×Q16—61no—
128C2×Q16—61no—
129C22×C8—31no—
130C22×C8—31no—
131C22×C8—31no—
132C22×C8—31no—
133C4⋊Q8—34no—
134C4⋊Q8—62no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—32no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8—62no—
141C4×Q8—62no—
142C4×Q8—121no—
143C4×Q8—121no—
144C4×Q8C4×D312yes—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C42—31no—
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—61no—
160C2×C42—61no—
161C2×C42—61no—
162#107—62no—
163#106—62no—
164C4⋊C8—34no—
165#103—32no—
166#103—34no—
167#103—34no—
168#103—32no—
169#103—62no—
170#103—62no—
171#103—61no—
172#103—61no—
173#103—121no—
174#103—121no—
175#97—62no—
176#97—62no—
177C4×C8—62no—
178#95—31no—
179#95—61no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95S411yescharacteristic
186C23×C3—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208SL(2,𝔽3)C4×D411yescharacteristic
209C2×Q8—31no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—32no—
213C2×Q8—32no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8C2×C4×D311yescharacteristic
225C2×Q8D3×D411yescharacteristic
226C2×Q8#23911yescharacteristic
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4C2×S411yescharacteristic
236C22×C4C2×S411yescharacteristic
237C22×C4C2×S411yescharacteristic
238Q16—62no—
239Q16—62no—
240Q16—62no—
241Q16—62no—
242Q16—62no—
243Q16—62no—
244Q16—62no—
245Q16—62no—
246C2×C8—32no—
247C2×C8—32no—
248C2×C8—32no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C2×C8—61no—
258C4⋊C4—32no—
259C4⋊C4—32no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—64no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—64no—
270C4⋊C4—62no—
271C4⋊C4—121no—
272C4⋊C4—122no—
273C4⋊C4—122no—
274C42—32no—
275C42—32no—
276C42—61no—
277C42—61no—
278C42—61no—
279C42—61no—
280C42—122no—
281C42—122no—
282C42—122no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C3×C4—82no—
291C3×C4—82no—
292C3×C4—161no—
293Dic3—82no—
294Dic3—82no—
295Dic3—82no—
296Dic3—161no—
297C23C22×S411yescharacteristic
298Q8—31no—
299Q8—31no—
300Q8—31no—
301Q8—61no—
302Q8—62no—
303Q8—62no—
304Q8—62no—
305Q8—62no—
306Q8—121no—
307Q8—121no—
308Q8—121no—
309Q8—121no—
310Q8#89311yescharacteristic
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—32no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—62no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4C4×S412yes—
346C2×C4#98412yes—
347C8—62no—
348C8—62no—
349C8—62no—
350C8—121no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C22C2×C4×S411yescharacteristic
359C22D4×S411yescharacteristic
360C22#591911yescharacteristic
361C22C2×#98411yescharacteristic
362C22#592311yescharacteristic
363C22#592711yescharacteristic
364C22#592711yescharacteristic
365C4—22no—
366C4—22no—
367C4—31no—
368C4—31no—
369C4—31no—
370C4—31no—
371C4—41no—
372C4—62no—
373C4—122no—
374C4—122no—
375C4—121no—
376C4—121no—
377C4—122no—
378C4—241no—
379C3—41noSylow
380C2#8530811yescharacteristic
381C2C4×#98411yescharacteristic
382C2#8535011yescharacteristic
383C2#8537611yescharacteristic
384C2#8538311yescharacteristic
385C2#8541711yescharacteristic
386C2#8542511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2351D3
prime radical, cosocleC22×SL(2,𝔽3)C23
hypercenter#95S4
hypocenterSL(2,𝔽3)C4×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
16. C4⋊#226C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#2262
C2×#53882
C2×C4×#2262
#2351⋊C32
#730632
#730612
#230143
#46544