Finite Group Databasebeta

← #11713888

Subgroups — #11713888

Order 768.

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

normal subgroups only

This group has 412 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
1C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C4⋊#859C211yesmaximal, characteristic
3#73059C211yesmaximal, characteristic
4#73059C211yesmaximal, characteristic
5#73059C211yesmaximal, characteristic
6#73059C211yesmaximal, characteristic
7C4×#859C211yesmaximal, characteristic
8#27298—31nomaximal, Sylow
9C2×#889—41nomaximal
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#2506—31no—
18Q8×C4⋊C4D311yescharacteristic
19#2100—31no—
20#2088—31no—
21#2084—31no—
22#2084—31no—
23#1941—31no—
24#1920—31no—
25#1916—31no—
26#1865—31no—
27#1865—31no—
28#1865—31no—
29#1865—31no—
30#1803—31no—
31C4×#103—31no—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×C3×C4⋊C4—41no—
34C2×C4⋊Dic3—41no—
35C2×Dic3⋊C4—41no—
36C2×Dic3⋊C4—41no—
37C2×Dic3⋊C4—41no—
38C2×Dic3⋊C4—41no—
39C2×C4×Dic3—41no—
40#889—48no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)—21no—
44#859—21no—
45#859—21no—
46#859—21no—
47#859Q812yes—
48C2×#125—61no—
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×#106—31no—
53C2×#106—31no—
54C2×#106—31no—
55C2×#106—31no—
56C2×C4⋊C8—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#103—31no—
60C2×#103—31no—
61C2×C4×C8—31no—
62#400—61no—
63#398—61no—
64#389—31no—
65#389—31no—
66#389—31no—
67#389—31no—
68#389—31no—
69#389—31no—
70#387—61no—
71#384—31no—
72#384—31no—
73C4×C4⋊C4—31no—
74C4×C4⋊C4—31no—
75#340—31no—
76#340—31no—
77#340—31no—
78#340—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#336—31no—
84#336—31no—
85#336—31no—
86#336—31no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×Dic3—41no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C2×SL(2,𝔽3)C2×Q811yescharacteristic
95C2×SL(2,𝔽3)#4111yescharacteristic
96C2×SL(2,𝔽3)#4111yescharacteristic
97C3×C4⋊C4—44no—
98C4⋊Dic3—44no—
99Dic3⋊C4—44no—
100Dic3⋊C4—44no—
101Dic3⋊C4—44no—
102Dic3⋊C4—44no—
103C4×Dic3—44no—
104C22×Q8C22×D311yescharacteristic
105C22×C8—31no—
106C22×C8—31no—
107C22×C8—31no—
108C22×C8—31no—
109#125—124no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8—61no—
114C4×Q8—61no—
115C4×Q8—61no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4S411yescharacteristic
132C2×C42—31no—
133C2×C42—31no—
134C2×C42—31no—
135C2×C42—61no—
136#106—62no—
137#106—62no—
138#106—121no—
139#106—121no—
140C4⋊C8—121no—
141#103—32no—
142#103—32no—
143#103—61no—
144#103—61no—
145#103—61no—
146#103—61no—
147#103—61no—
148#103—61no—
149C4×C8—121no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—41no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×C3×C4—81no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4—81no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—42no—
173C2×Dic3—81no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176C2×Dic3—81no—
177C2×Dic3—81no—
178C2×Dic3—81no—
179SL(2,𝔽3)#12511yescharacteristic, complemented
180C2×Q8—31no—
181C2×Q8—31no—
182C2×Q8—31no—
183C2×Q8#23511yescharacteristic
184C2×Q8#23711yescharacteristic
185C2×Q8D3×Q811yescharacteristic
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—61no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—61no—
194C22×C4C2×S411yescharacteristic
195C22×C4C2×S411yescharacteristic
196C22×C4C2×S411yescharacteristic
197C2×C8—32no—
198C2×C8—61no—
199C2×C8—61no—
200C2×C8—61no—
201C2×C8—61no—
202C2×C8—61no—
203C2×C8—61no—
204C2×C8—61no—
205C2×C8—61no—
206C2×C8—61no—
207C2×C8—61no—
208C2×C8—61no—
209C4⋊C4—32no—
210C4⋊C4—32no—
211C4⋊C4—61no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—121no—
224C4⋊C4—122no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C4⋊C4—122no—
228C4⋊C4—122no—
229C4⋊C4—122no—
230C4⋊C4—241no—
231C4⋊C4GL(2,𝔽3)14yes—
232C42—61no—
233C42—61no—
234C42—61no—
235C42—61no—
236C42—61no—
237C42—61no—
238C42—122no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C3×C4—82no—
247C3×C4—82no—
248C3×C4—82no—
249Dic3—44no—
250Dic3—82no—
251Dic3—82no—
252Dic3—82no—
253C23C22×S411yescharacteristic
254Q8—31no—
255Q8—31no—
256Q8—31no—
257Q8—61no—
258Q8#88911yescharacteristic, complemented
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—62no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4C2×GL(2,𝔽3)12yes—
293C2×C4C2×GL(2,𝔽3)12yes—
294C2×C4C2×GL(2,𝔽3)12yes—
295C8—62no—
296C8—121no—
297C8—121no—
298C8—121no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C22#591211yescharacteristic
307C22#591411yescharacteristic
308C22C22×GL(2,𝔽3)11yescharacteristic
309C22Q8×S411yescharacteristic
310C22#592311yescharacteristic
311C22#592411yescharacteristic
312C22#592711yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—31no—
317C4—31no—
318C4—31no—
319C4—31no—
320C4—121no—
321C4—121no—
322C4—121no—
323C4—122no—
324C4—241no—
325C4—241no—
326C4—241no—
327C3—41noSylow
328C2#8530411yescharacteristic
329C2#8532511yescharacteristic
330C2#8536211yescharacteristic
331C2#8539711yescharacteristic
332C2#8540511yescharacteristic
333C2#8541611yescharacteristic
334C2Q8×GL(2,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C4⋊C4D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#125
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
179. SL(2,𝔽3)#125semidirect
258. Q8#889semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×SL(2,𝔽3)2
C4⋊#8592
#730592
C4×#8592
#272983
C2×#8894