Finite Group Databasebeta

← #11714011

Subgroups — #11714011

Order 768.

551 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 71 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×#41⋊C3C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal
4#73062C212yesmaximal, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#32955—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C4×#41⋊C3—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C2×#859C2212yes—
13#5386—22no—
14#4929—41nomaximal
15C2×C4×#41D311yescharacteristic
16C2×#475—31no—
17C2×#474—31no—
18#2857—31no—
19#1864—32no—
20#1863—32no—
21#1854—32no—
22#1803—31no—
23#1802—31no—
24#1797—31no—
25#1788—32no—
26C22×Dic6—41no—
27C2×#41⋊C3—21no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3Q812yescomplemented
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×C3×C42—41no—
32C2×C4⋊Dic3—41no—
33C4×SL(2,𝔽3)—21no—
34C4×SL(2,𝔽3)D412yes—
35#859—22no—
36#859—22no—
37C3⋊#95—42no—
38C3⋊#95—42no—
39C22×#41C2×D311yescharacteristic
40C2×C4⋊Q8—31no—
41C2×C22⋊Q8—61no—
42C4×#41—22no—
43C4×#41—62no—
44C2×C4×Q8C2×D311yescharacteristic
45C2×C4×Q8C2×D311yescharacteristic
46C2×C4×D4—31no—
47C2×C4×D4—31no—
48C2×#117—31no—
49C22×C42—31no—
50#475—62no—
51#475—62no—
52#474—64no—
53C2×#107—32no—
54C2×#106—32no—
55C2×C4⋊C8—31no—
56C2×C4⋊C8—31no—
57C2×#103—32no—
58C2×#103—32no—
59C2×#102—32no—
60C2×#102—32no—
61C2×C22⋊C8—32no—
62#391—61no—
63#389—32no—
64#389—61no—
65#384—31no—
66#380—32no—
67#380—61no—
68#340—32no—
69#340—32no—
70#336—32no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76C2×Dic6—44no—
77C2×Dic6—82no—
78C2×Dic6—82no—
79#41⋊C3—22no—
80C2×SL(2,𝔽3)C2×D411yescharacteristic
81C2×SL(2,𝔽3)C2×Q811yescharacteristic
82C2×SL(2,𝔽3)#4111yescharacteristic
83C3×C42—82no—
84C4⋊Dic3—82no—
85C2×#41—21no—
86C2×#41—21no—
87C2×#41—32no—
88C2×#41—61no—
89C2×#41—61no—
90C2×#41Dic612yescomplemented
91C22×Q8—61no—
92C22×Q8C22×D311yescharacteristic
93C22×D4—31no—
94C23×C4—31no—
95C23×C4—31no—
96C23×C4—31no—
97C22×C8—32no—
98C22×C8—32no—
99C4⋊Q8—62no—
100C4⋊Q8—62no—
101C22⋊Q8—124no—
102C4×Q8—21no—
103C4×Q8—32no—
104C4×Q8—61no—
105C4×Q8#6712yes—
106C4×D4—34no—
107C4×D4—62no—
108C4×D4—62no—
109C4×D4—62no—
110#117—62no—
111#117—62no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—61no—
120C2×C22⋊C4—61no—
121C2×C22⋊C4—62no—
122C2×C42—31no—
123C2×C42—31no—
124C2×C42—31no—
125C2×C42—62no—
126C2×C42—62no—
127C2×C42S411yescharacteristic
128#107—64no—
129#106—64no—
130C4⋊C8—62no—
131C4⋊C8—62no—
132#103—62no—
133#103—62no—
134#103—62no—
135#103—62no—
136#102—64no—
137#102—64no—
138C22⋊C8—64no—
139#95—62no—
140#95—62no—
141C23×C3—41no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—81no—
148C2×C3×C4—81no—
149C2×C3×C4—81no—
150C2×C3×C4—81no—
151C2×Dic3—44no—
152C2×Dic3—82no—
153C2×Dic3—82no—
154C2×Dic3—82no—
155C2×Dic3—82no—
156C2×Dic3—82no—
157Dic6—88no—
158SL(2,𝔽3)C22⋊Q811yescharacteristic, complemented
159C24—31no—
160#41—22no—
161#41—64no—
162#41—62no—
163#41—62no—
164#41—62no—
165C2×Q8—31no—
166C2×Q8—31no—
167C2×Q8—31no—
168C2×Q8—122no—
169C2×Q8—122no—
170C2×Q8—122no—
171C2×Q8C2×Dic611yescharacteristic
172C2×Q8#23511yescharacteristic
173C2×Q8C2×#6711yescharacteristic
174C2×D4—32no—
175C2×D4—32no—
176C2×D4—32no—
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—61no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—32no—
184C22×C4—31no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4—61no—
188C22×C4—61no—
189C22×C4—62no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—121no—
197C22×C4C2×S411yescharacteristic
198C22×C4C2×S411yescharacteristic
199C22×C4C2×S411yescharacteristic
200C2×C8—62no—
201C2×C8—62no—
202C2×C8—62no—
203C2×C8—62no—
204C2×C8—62no—
205C2×C8—62no—
206C4⋊C4—32no—
207C4⋊C4—32no—
208C4⋊C4—61no—
209C4⋊C4—61no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4—122no—
215C4⋊C4—122no—
216C4⋊C4—124no—
217C4⋊C4—122no—
218C22⋊C4—64no—
219C22⋊C4—124no—
220C42—22no—
221C42—32no—
222C42—32no—
223C42—62no—
224C42—61no—
225C42—61no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—41no—
233C3×C4—44no—
234C3×C4—82no—
235C3×C4—82no—
236Dic3—84no—
237Dic3—162no—
238C23—32no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23—61no—
245C23C22×S411yescharacteristic
246Q8—31no—
247Q8—31no—
248Q8—31no—
249Q8—61no—
250Q8—244no—
251Q8#92411yescharacteristic
252D4—34no—
253D4—62no—
254D4—62no—
255D4—62no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—32no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—124no—
277C2×C4—124no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4A4⋊Q812yes—
285C2×C4#98412yes—
286C2×C4#98512yes—
287C2×C4#98612yes—
288C2×C4GL(2,ℤ/4)12yes—
289C8—122no—
290C8—122no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C22—62no—
299C22—62no—
300C22—62no—
301C22—121no—
302C22—121no—
303C22—121no—
304C22—121no—
305C22C2×A4⋊Q811yescharacteristic
306C22#591211yescharacteristic
307C22C2×#98411yescharacteristic
308C22C2×#98511yescharacteristic
309C22C2×#98611yescharacteristic
310C22#592311yescharacteristic
311C22C2×GL(2,ℤ/4)11yescharacteristic
312C4—22no—
313C4—22no—
314C4—31no—
315C4—31no—
316C4—31no—
317C4—31no—
318C4—62no—
319C4—121no—
320C4—242no—
321C4—242no—
322C4#542714yes—
323C3—41noSylow
324C2—122no—
325C2#8533411yescharacteristic
326C2#8533511yescharacteristic
327C2#8533711yescharacteristic
328C2#8534811yescharacteristic
329C2#8548011yescharacteristic
330C2#8551711yescharacteristic
331C2C2×#542711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×C4×#41D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#5386C2semidirect
4. #73062C2semidirect
5. C4⋊#859C2semidirect
29. C2×#41⋊C3Q8semidirect
90. C2×#41Dic6semidirect
158. SL(2,𝔽3)C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#41⋊C32
C2×#53862
#731512
#730622
C4⋊#8592
#329553
#49294