Finite Group Databasebeta

← #11713973

Subgroups — #11713973

Order 768.

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

normal subgroups only

This group has 605 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×#5425C212yesmaximal
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4#73061C212yesmaximal, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#32907—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×#226C2212yes—
9C4×#41⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12#5425—24no—
13C2×#859C2212yes—
14C4⋊#226—22no—
15#4928—41nomaximal
16C2×C4×#41D311yescharacteristic
17C2×#462—31no—
18C2×#461—31no—
19C2×#452—32no—
20#2857—31no—
21#1878—32no—
22#1877—32no—
23#1873—32no—
24#1803—31no—
25#1802—31no—
26#1797—31no—
27C2×#41⋊C3—21no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3D412yes—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×#226—22no—
32C2×#226—22no—
33C2×#226—22no—
34C2×C3×C42—41no—
35C2×C4⋊Dic3—41no—
36C2×C4⋊Dic3—41no—
37C2×Dic3⋊C4—42no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)D412yes—
40#859—22no—
41C3⋊#95—42no—
42C22×#41C2×D311yescharacteristic
43C22×Q16—32no—
44C22×SD16—32no—
45C2×C4⋊Q8—31no—
46C2×C22⋊Q8—32no—
47C4×#41—22no—
48C4×#41—62no—
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×D4—31no—
52C2×C4×D4—31no—
53C2×#117—31no—
54C22×C42—31no—
55#462—62no—
56#462—62no—
57#461—64no—
58#452—64no—
59#452—64no—
60C2×C4⋊C8—31no—
61C2×C4⋊C8—31no—
62C2×#103—32no—
63C2×#103—32no—
64C2×#103—32no—
65C2×#102—32no—
66C2×C22⋊C8—32no—
67#391—32no—
68#389—61no—
69#389—61no—
70#384—31no—
71#380—61no—
72#380—61no—
73#340—32no—
74#340—32no—
75#336—32no—
76C22×C3×C4—41no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×Dic3—42no—
80C22×Dic3—42no—
81#41⋊C3—22no—
82C2×SL(2,𝔽3)C2×D411yescharacteristic
83C2×SL(2,𝔽3)C2×D411yescharacteristic
84C2×SL(2,𝔽3)#4111yescharacteristic
85#226—42no—
86C3×C42—82no—
87C4⋊Dic3—44no—
88C4⋊Dic3—82no—
89Dic3⋊C4—84no—
90C2×#41—21no—
91C2×#41—21no—
92C2×#41—32no—
93C2×#41—61no—
94C2×#41—61no—
95C2×#41D1212yes—
96C22×Q8—32no—
97C22×Q8C22×D311yescharacteristic
98C22×D4—31no—
99C23×C4—31no—
100C23×C4—31no—
101C23×C4—31no—
102C2×Q16—62no—
103C2×Q16—62no—
104C2×Q16—62no—
105C2×Q16—62no—
106C2×Q16—62no—
107C2×Q16—62no—
108C2×SD16—64no—
109C2×SD16—64no—
110C2×SD16—64no—
111C22×C8—32no—
112C22×C8—32no—
113C4⋊Q8—62no—
114C4⋊Q8—62no—
115C22⋊Q8—64no—
116C22⋊Q8—64no—
117C4×Q8—21no—
118C4×Q8—32no—
119C4×Q8—61no—
120C4×Q8#6712yes—
121C4×D4—34no—
122C4×D4—62no—
123C4×D4—62no—
124C4×D4—62no—
125#117—62no—
126#117—62no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C22⋊C4—61no—
136C2×C22⋊C4—62no—
137C2×C42—31no—
138C2×C42—31no—
139C2×C42—31no—
140C2×C42—62no—
141C2×C42—62no—
142C2×C42S411yescharacteristic
143C4⋊C8—62no—
144C4⋊C8—62no—
145#103—64no—
146#103—64no—
147#103—62no—
148#103—62no—
149#102—64no—
150C22⋊C8—64no—
151#95—62no—
152#95—62no—
153C23×C3—41no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—81no—
160C2×C3×C4—81no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×Dic3—44no—
164C2×Dic3—82no—
165C2×Dic3—82no—
166C2×Dic3—82no—
167C2×Dic3—82no—
168C2×Dic3—82no—
169SL(2,𝔽3)#12111yescharacteristic
170C24—31no—
171#41—22no—
172#41—64no—
173#41—62no—
174#41—62no—
175#41—62no—
176C2×Q8—31no—
177C2×Q8—31no—
178C2×Q8—31no—
179C2×Q8—62no—
180C2×Q8—62no—
181C2×Q8—62no—
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8—62no—
185C2×Q8C2×D1211yescharacteristic
186C2×Q8#23511yescharacteristic
187C2×Q8C2×#6711yescharacteristic
188C2×D4—32no—
189C2×D4—32no—
190C2×D4—32no—
191C2×D4—61no—
192C2×D4—61no—
193C2×D4—61no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—32no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4—61no—
202C22×C4—61no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—121no—
211C22×C4C2×S411yescharacteristic
212C22×C4C2×S411yescharacteristic
213C22×C4C2×S411yescharacteristic
214Q16—122no—
215Q16—122no—
216Q16—122no—
217Q16—122no—
218SD16—128no—
219C2×C8—62no—
220C2×C8—62no—
221C2×C8—62no—
222C2×C8—62no—
223C2×C8—62no—
224C2×C8—62no—
225C4⋊C4—32no—
226C4⋊C4—32no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—62no—
230C4⋊C4—62no—
231C4⋊C4—122no—
232C4⋊C4—122no—
233C4⋊C4—124no—
234C4⋊C4—122no—
235C4⋊C4—122no—
236C22⋊C4—64no—
237C22⋊C4—124no—
238C42—22no—
239C42—32no—
240C42—32no—
241C42—62no—
242C42—61no—
243C42—61no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C3×C4—44no—
252C3×C4—82no—
253C3×C4—82no—
254Dic3—84no—
255Dic3—162no—
256C23—32no—
257C23—61no—
258C23—61no—
259C23—61no—
260C23—61no—
261C23—61no—
262C23—61no—
263C23C22×S411yescharacteristic
264Q8—31no—
265Q8—31no—
266Q8—31no—
267Q8—61no—
268Q8—122no—
269Q8—122no—
270Q8—122no—
271Q8—122no—
272Q8#93011yescharacteristic
273D4—34no—
274D4—62no—
275D4—62no—
276D4—62no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—124no—
298C2×C4—124no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4C4⋊S412yes—
306C2×C4#98412yes—
307C2×C4#98412yes—
308C2×C4#98512yes—
309C2×C4GL(2,ℤ/4)12yes—
310C8—122no—
311C8—122no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C22—62no—
320C22—62no—
321C22—62no—
322C22—121no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22C2×C4⋊S411yescharacteristic
327C22#591211yescharacteristic
328C22C2×#98411yescharacteristic
329C22C2×#98411yescharacteristic
330C22C2×#98511yescharacteristic
331C22#592311yescharacteristic
332C22C2×GL(2,ℤ/4)11yescharacteristic
333C4—22no—
334C4—22no—
335C4—31no—
336C4—31no—
337C4—31no—
338C4—31no—
339C4—62no—
340C4—121no—
341C4—242no—
342C4—242no—
343C4#542514yescomplemented
344C3—41noSylow
345C2—122no—
346C2#8533511yescharacteristic
347C2#8533611yescharacteristic
348C2#8533811yescharacteristic
349C2#8534411yescharacteristic
350C2#8548611yescharacteristic
351C2C2×#542511yescharacteristic
352C2#8551711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×C4×#41D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#121
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
3. C2×C4⋊#226C2semidirect
4. #73061C2semidirect
5. C4⋊#859C2semidirect
343. C4#5425semidirect

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×#54252
C2×C4⋊#2262
#730612
C4⋊#8592
#329073
#49284