Finite Group Databasebeta

← #11713894

Subgroups — #11713894

Order 768.

371 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 55 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
2#73063C211yesmaximal, characteristic
3#73062C211yesmaximal, characteristic
4#73062C211yesmaximal, characteristic
5C4⋊#859C211yesmaximal, characteristic
6#73059C211yesmaximal, characteristic
7#73059C211yesmaximal, characteristic
8#27311—31nomaximal, Sylow
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#4981—41nomaximal
17#2506—31no—
18Q8×C4⋊C4D311yescharacteristic
19#2099—31no—
20#2087—31no—
21#2084—31no—
22#2083—31no—
23#1944—31no—
24#1919—31no—
25#1917—31no—
26#1865—31no—
27#1865—31no—
28#1863—31no—
29#1863—31no—
30#1803—31no—
31#1763—31no—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×C3×C4⋊C4—41no—
34C2×C4⋊Dic3—41no—
35C2×Dic3⋊C4—41no—
36C2×Dic3⋊C4—41no—
37C4×SL(2,𝔽3)—21no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—21no—
40#859—21no—
41#859—21no—
42#859—21no—
43#859Q812yes—
44C3⋊#95—41no—
45C3⋊#95—41no—
46C3⋊#95—41no—
47C2×#125—61no—
48C2×C4×Q8C2×D311yescharacteristic
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×#107—31no—
52C2×#107—31no—
53C2×#106—31no—
54C2×#106—31no—
55C2×C4⋊C8—31no—
56C2×#103—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#103—31no—
60C2×#97—31no—
61#400—61no—
62#398—61no—
63#389—31no—
64#389—31no—
65#389—31no—
66#389—31no—
67#389—31no—
68#389—31no—
69#387—61no—
70#384—31no—
71#384—31no—
72C4×C4⋊C4—31no—
73C4×C4⋊C4—31no—
74#340—31no—
75#340—31no—
76#340—31no—
77#340—31no—
78#340—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#336—31no—
83#336—31no—
84#336—31no—
85#336—31no—
86C22×C3×C4—41no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×Dic3—41no—
90C22×Dic3—41no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C2×SL(2,𝔽3)C2×Q811yescharacteristic
94C2×SL(2,𝔽3)#4111yescharacteristic
95C2×SL(2,𝔽3)#4111yescharacteristic
96C3×C4⋊C4—82no—
97C4⋊Dic3—82no—
98Dic3⋊C4—82no—
99Dic3⋊C4—82no—
100C22×Q8C22×D311yescharacteristic
101C22×C8—31no—
102C22×C8—31no—
103C22×C8—31no—
104C22×C8—31no—
105#125—124no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×Q8—61no—
110C4×Q8—61no—
111C4×Q8—61no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4S411yescharacteristic
128C2×C42—31no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—61no—
132#107—121no—
133#107—121no—
134#106—62no—
135#106—62no—
136C4⋊C8—121no—
137#103—32no—
138#103—32no—
139#103—61no—
140#103—61no—
141#103—61no—
142#103—61no—
143#103—61no—
144#103—61no—
145#97—121no—
146#95—31no—
147#95—31no—
148#95—61no—
149#95—61no—
150#95—61no—
151#95—61no—
152#95—61no—
153C23×C3—41no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—81no—
157C2×C3×C4—81no—
158C2×C3×C4—81no—
159C2×C3×C4—81no—
160C2×C3×C4—81no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—81no—
166C2×Dic3—81no—
167C2×Dic3—81no—
168C2×Dic3—81no—
169C2×Dic3—81no—
170C2×Dic3—81no—
171C2×Dic3—81no—
172C2×Dic3—81no—
173C2×Dic3—81no—
174C2×Dic3—81no—
175SL(2,𝔽3)#12511yescharacteristic, complemented
176C2×Q8—31no—
177C2×Q8—31no—
178C2×Q8—31no—
179C2×Q8#23511yescharacteristic
180C2×Q8#23711yescharacteristic
181C2×Q8D3×Q811yescharacteristic
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—61no—
187C22×C4—61no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4C2×S411yescharacteristic
191C22×C4C2×S411yescharacteristic
192C22×C4C2×S411yescharacteristic
193C2×C8—32no—
194C2×C8—61no—
195C2×C8—61no—
196C2×C8—61no—
197C2×C8—61no—
198C2×C8—61no—
199C2×C8—61no—
200C2×C8—61no—
201C2×C8—61no—
202C2×C8—61no—
203C2×C8—61no—
204C2×C8—61no—
205C4⋊C4—22no—
206C4⋊C4—32no—
207C4⋊C4—32no—
208C4⋊C4—61no—
209C4⋊C4—61no—
210C4⋊C4—61no—
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—62no—
221C4⋊C4—122no—
222C4⋊C4—122no—
223C4⋊C4—122no—
224C4⋊C4—122no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C4⋊C4—241no—
228C42—61no—
229C42—61no—
230C42—61no—
231C42—61no—
232C42—61no—
233C42—61no—
234C42—122no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C3×C4—82no—
243C3×C4—82no—
244C3×C4—161no—
245Dic3—82no—
246Dic3—82no—
247Dic3—161no—
248Dic3—161no—
249C23C22×S411yescharacteristic
250Q8—31no—
251Q8—31no—
252Q8—31no—
253Q8—61no—
254Q8#88911yescharacteristic
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
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#98312yes—
290C2×C4#98612yes—
291C8—62no—
292C8—121no—
293C8—121no—
294C8—121no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C22#591211yescharacteristic
303C22#591411yescharacteristic
304C22C2×#98311yescharacteristic
305C22Q8×S411yescharacteristic
306C22C2×#98611yescharacteristic
307C22#592411yescharacteristic
308C22#592711yescharacteristic
309C4—22no—
310C4—22no—
311C4—31no—
312C4—31no—
313C4—31no—
314C4—31no—
315C4—41no—
316C4—121no—
317C4—121no—
318C4—121no—
319C4—122no—
320C4—241no—
321C4—241no—
322C4—241no—
323C3—41noSylow
324C2#8530411yescharacteristic
325C2#8533211yescharacteristic
326C2#8536211yescharacteristic
327C2#8539411yescharacteristic
328C2#8541211yescharacteristic
329C2#8542411yescharacteristic
330C2#8542611yescharacteristic
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
175. SL(2,𝔽3)#125semidirect

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
#730632
#730622
C4⋊#8592
#730592
#273113
#49814