Finite Group Databasebeta

← #11713679

Subgroups — #11713679

Order 768.

451 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 61 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×#557⋊C3C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic
3#73062C211yesmaximal, characteristic
4#73062C211yesmaximal, characteristic
5#73059C211yesmaximal, characteristic
6#73059C211yesmaximal, characteristic
7C4×#859C211yesmaximal, characteristic
8#25842—31nomaximal, Sylow
9#557⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5389—22no—
18#4976—41nomaximal
19C2×#557D311yescharacteristic
20C2×#499—31no—
21C2×#479—31no—
22#2584—31no—
23#2059—31no—
24#2059—31no—
25#2058—31no—
26#2057—31no—
27#1989—31no—
28#1984—31no—
29#1865—31no—
30#1865—31no—
31#1863—31no—
32#1863—31no—
33C4×#103—31no—
34C22×SL(2,𝔽3)C2311yescharacteristic
35C2×C3×C4⋊C4—41no—
36C2×Dic3⋊C4—41no—
37C2×Dic3⋊C4—41no—
38C2×Dic3⋊C4—41no—
39C2×C4×Dic3—41no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)D412yes—
43#859—21no—
44#859—21no—
45#859Q812yescomplemented
46#859Q812yescomplemented
47C3⋊#95—41no—
48C3⋊#95—41no—
49#557—22no—
50#557—62no—
51C2×C4⋊Q8—31no—
52C2×#125—31no—
53C2×#125—31no—
54C2×#125—31no—
55C2×C4×Q8—31no—
56C2×C4×Q8C2×D311yescharacteristic
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×Q8C2×D311yescharacteristic
59#499—64no—
60#479—62no—
61#479—62no—
62C2×#107—31no—
63C2×#107—31no—
64C2×#107—31no—
65C2×#106—31no—
66C2×#106—31no—
67C2×#106—31no—
68C2×C4⋊C8—31no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×C4×C8—31no—
74#398—61no—
75#395—61no—
76#389—31no—
77#389—31no—
78#389—31no—
79#389—31no—
80#389—61no—
81#389—61no—
82C4×C4⋊C4—31no—
83#340—31no—
84#340—31no—
85#340—31no—
86#340—31no—
87#340—31no—
88#340—31no—
89#336—31no—
90#336—31no—
91#336—31no—
92#336—31no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C2×SL(2,𝔽3)C2×D411yescharacteristic
101C2×SL(2,𝔽3)C2×Q811yescharacteristic
102C2×SL(2,𝔽3)C2×Q811yescharacteristic
103C3×C4⋊C4—82no—
104Dic3⋊C4—44no—
105Dic3⋊C4—82no—
106Dic3⋊C4—82no—
107C4×Dic3—82no—
108C22×Q8—31no—
109C22×Q8C22×D311yescharacteristic
110C22×C8—31no—
111C22×C8—31no—
112C22×C8—31no—
113C22×C8—31no—
114C4⋊Q8—62no—
115C4⋊Q8—62no—
116#125—34no—
117#125—62no—
118#125—62no—
119#125—62no—
120#125—62no—
121#125—62no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—32no—
125C4×Q8—62no—
126C4×Q8—62no—
127C4×Q8—61no—
128C4×Q8—61no—
129C4×Q8#6712yes—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4S411yescharacteristic
147C2×C42—31no—
148C2×C42—31no—
149C2×C42—31no—
150C2×C42—61no—
151#107—62no—
152#107—62no—
153#107—62no—
154#106—62no—
155#106—62no—
156#106—62no—
157C4⋊C8—62no—
158#103—32no—
159#103—32no—
160#103—32no—
161#103—32no—
162#103—61no—
163#103—61no—
164#103—61no—
165#103—61no—
166C4×C8—62no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172C23×C3—41no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—81no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194SL(2,𝔽3)C4⋊Q811yescharacteristic
195C2×Q8—31no—
196C2×Q8—31no—
197C2×Q8—31no—
198C2×Q8—62no—
199C2×Q8—62no—
200C2×Q8—121no—
201C2×Q8D3×Q811yescharacteristic
202C2×Q8D3×Q811yescharacteristic
203C2×Q8C2×#6711yescharacteristic
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4C2×S411yescharacteristic
213C22×C4C2×S411yescharacteristic
214C22×C4C2×S411yescharacteristic
215C2×C8—32no—
216C2×C8—32no—
217C2×C8—61no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C4⋊C4—22no—
228C4⋊C4—32no—
229C4⋊C4—32no—
230C4⋊C4—32no—
231C4⋊C4—32no—
232C4⋊C4—32no—
233C4⋊C4—62no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—64no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C4⋊C4—122no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—241no—
250C4⋊C4—241no—
251C42—32no—
252C42—32no—
253C42—61no—
254C42—61no—
255C42—61no—
256C42—61no—
257C42—241no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C3×C4—82no—
266C3×C4—82no—
267C3×C4—161no—
268Dic3—44no—
269Dic3—82no—
270Dic3—82no—
271Dic3—161no—
272C23C22×S411yescharacteristic
273Q8—31no—
274Q8—31no—
275Q8—31no—
276Q8—61no—
277Q8—124no—
278Q8#94411yescharacteristic
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—32no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4#98512yes—
314C2×C4GL(2,ℤ/4)12yes—
315C8—62no—
316C8—62no—
317C8—121no—
318C8—121no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—41no—
326C22Q8×S411yescharacteristic
327C22Q8×S411yescharacteristic
328C22C2×#98511yescharacteristic
329C22#592311yescharacteristic
330C22#592611yescharacteristic
331C22#592711yescharacteristic
332C22C2×GL(2,ℤ/4)11yescharacteristic
333C4—22no—
334C4—22no—
335C4—31no—
336C4—31no—
337C4—31no—
338C4—31no—
339C4—41no—
340C4—62no—
341C4—121no—
342C4—121no—
343C4—122no—
344C4—122no—
345C4—241no—
346C4—241no—
347C3—41noSylow
348C2#8541411yescharacteristic
349C2#8541611yescharacteristic
350C2#8541811yescharacteristic
351C2#8542311yescharacteristic
352C2#8550211yescharacteristic
353C2#8552111yescharacteristic
354C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
45. #859Q8semidirect
46. #859Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×#53892
#730622
#730592
C4×#8592
#258423
#49764