Finite Group Databasebeta

← #11713023

Subgroups — #11713023

Order 768.

468 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 65 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×#5405C212yesmaximal
2C4×#985C211yesmaximal, characteristic
3C4⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#73122C212yesmaximal, complemented
5C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
6#22686—31nomaximal, Sylow
7C2×#985C2211yescharacteristic
8C2×#131⋊C3C2212yes—
9C4×#41⋊C3C2211yescharacteristic
10#5424C2211yescharacteristic
11C2×GU(2,𝔽3)C2212yes—
12#5405—24no—
13C4×GL(2,𝔽3)C411yescharacteristic, complemented
14C4×GL(2,𝔽3)C411yescharacteristic, complemented
15C4×#226C411yescharacteristic, complemented
16C4×#226C411yescharacteristic, complemented
17#4837—41nomaximal
18C2×#444—32no—
19C4×#135—31no—
20#2906—31no—
21#2906D311yescharacteristic, complemented
22#1946—32no—
23#1866—32no—
24#1866—32no—
25#1848—31no—
26#1806—31no—
27#1806—31no—
28#1770—31no—
29C2×#41⋊C3C2311yescharacteristic
30#985D412yes—
31#985Q812yes—
32C2×GL(2,𝔽3)C2×C411yescharacteristic
33C2×#226C2×C411yescharacteristic
34C2×ΓL(1,𝔽25)—42no—
35C42×D3—41no—
36#131⋊C3—22no—
37C4×SL(2,𝔽3)C2×C411yescharacteristic
38C4×SL(2,𝔽3)C2×C411yescharacteristic
39GU(2,𝔽3)—22no—
40#859C2×C411yescharacteristic
41#859C2×C411yescharacteristic
42C3×C4⋊C8—41no—
43Dic3⋊C8—42no—
44C4⋊(C3⋊C8)—41no—
45C2×#135—31no—
46C2×#131—32no—
47C2×#131C2×D312yes—
48C4×#41—31no—
49C4×#41C2×D311yescharacteristic
50#444—64no—
51#444—64no—
52C4×Q16—31no—
53C4×Q16—31no—
54C4×SD16—31no—
55C4×SD16—31no—
56C4×SD16—31no—
57C4×SD16—31no—
58C4×D8—31no—
59C4×D8—31no—
60C2×#108—32no—
61C2×#108—32no—
62#427—31no—
63#424—32no—
64#424—32no—
65#424—62no—
66#423—31no—
67#423—31no—
68#423—61no—
69C2×C4⋊C8—31no—
70C2×C4⋊C8—61no—
71C2×#104—32no—
72C2×#104—32no—
73#416—31no—
74#416—31no—
75C2×#97—32no—
76C2×C4×C8—31no—
77C4×C4⋊C4—31no—
78#339—32no—
79#339—32no—
80#336—32no—
81C2×C4×D3—41no—
82C2×C4×D3—41no—
83C2×C4×D3—41no—
84#41⋊C3C2×D411yescharacteristic
85#41⋊C3C2×Q811yescharacteristic
86C2×SL(2,𝔽3)C22×C411yescharacteristic
87GL(2,𝔽3)C4⋊C412yescomplemented
88#226C4⋊C412yescomplemented
89C2×C3×C8—42no—
90C3×C42—41no—
91C4×Dic3—41no—
92C4×Dic3—41no—
93C4×Dic3—41no—
94C2×C3⋊C8—42no—
95ΓL(1,𝔽25)—84no—
96C2×#41—31no—
97C2×#41C22×D311yescharacteristic
98#135—32no—
99#135—32no—
100#135—32no—
101#135—32no—
102C2×Q16—31no—
103C2×SD16—31no—
104C2×SD16—31no—
105C2×D8—31no—
106#131—22no—
107#131—62no—
108#131—64no—
109#131—122no—
110C2×M16—32no—
111C2×M16—32no—
112C2×M16—62no—
113C22×C8—31no—
114C22×C8—31no—
115C22×C8—32no—
116C22×C8—62no—
117C4×Q8—31no—
118C4×Q8—31no—
119C4×Q8C4×D311yescharacteristic
120C4×Q8C4×D311yescharacteristic
121C4×D4—31no—
122C4×D4—31no—
123C4×D4—31no—
124C4×D4—31no—
125C4×D4—61no—
126C4×D4—61no—
127#117—31no—
128#117—31no—
129#117—61no—
130C2×C4⋊C4—61no—
131C2×C42—31no—
132C2×C42—32no—
133C2×C42—61no—
134#108—64no—
135#108—122no—
136#107—31no—
137#107—31no—
138#106—31no—
139#106—31no—
140C4⋊C8—32no—
141C4⋊C8—31no—
142C4⋊C8—61no—
143C4⋊C8—61no—
144C4⋊C8—62no—
145C4⋊C8—62no—
146C4⋊C8—61no—
147C4⋊C8S411yescharacteristic
148#104—64no—
149#104—62no—
150#104—62no—
151#103—31no—
152#103—31no—
153#103—31no—
154#103—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#97—64no—
160C4×C8—31no—
161C4×C8—31no—
162C4×C8—31no—
163C4×C8—31no—
164#95—61no—
165C22×D3—41no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×Dic3—41no—
170C2×Dic3—41no—
171C2×Dic3—41no—
172C4×D3—42no—
173C4×D3—42no—
174C4×D3—82no—
175C4×D3—82no—
176SL(2,𝔽3)C2×C4⋊C411yescharacteristic
177C3×C8—82no—
178C3⋊C8—82no—
179#41—31no—
180#41—31no—
181#41—32no—
182#41—32no—
183#41—61no—
184#41—61no—
185#41D3×D411yescharacteristic
186#41D3×Q811yescharacteristic
187C2×Q8—31no—
188C2×Q8C2×C4×D311yescharacteristic
189C2×D4—31no—
190C2×D4—31no—
191C2×D4—61no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—62no—
199Q16—32no—
200Q16—32no—
201SD16—32no—
202SD16—32no—
203SD16—32no—
204SD16—32no—
205D8—32no—
206D8—32no—
207M16—62no—
208M16—62no—
209M16—62no—
210M16—62no—
211M16—124no—
212C2×C8—31no—
213C2×C8—31no—
214C2×C8—32no—
215C2×C8—32no—
216C2×C8—31no—
217C2×C8—31no—
218C2×C8—32no—
219C2×C8—62no—
220C2×C8—62no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—62no—
224C2×C8—124no—
225C2×C8C2×S412yes—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—122no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C42—31no—
238C42—31no—
239C42—31no—
240C42—62no—
241C42—62no—
242C42—61no—
243C42—61no—
244C42—61no—
245C42C2×S411yescharacteristic
246C22×C3—41no—
247C2×D3—42no—
248C2×D3—81no—
249C2×D3—81no—
250C3×C4—41no—
251C3×C4—42no—
252C3×C4—41no—
253C3×C4—81no—
254Dic3—42no—
255Dic3—42no—
256Dic3—81no—
257C23—31no—
258C23—61no—
259Q8—31no—
260Q8—32no—
261Q8—61no—
262Q8D3×C4⋊C411yescharacteristic
263D4—31no—
264D4—31no—
265D4—32no—
266D4—61no—
267D4—61no—
268D4—62no—
269D4—62no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—121no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—241no—
289C2×C4C4×S411yescharacteristic
290C2×C4C4×S411yescharacteristic
291C2×C4C22×S411yescharacteristic
292C8—22no—
293C8—32no—
294C8—32no—
295C8—61no—
296C8—61no—
297C8—62no—
298C8—122no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302D3—82no—
303C22—31no—
304C22—31no—
305C22—61no—
306C22—61no—
307C22—62no—
308C22—121no—
309C22—121no—
310C22C2×C4×S411yescharacteristic
311C4—21no—
312C4—31no—
313C4—31no—
314C4—62no—
315C4—61no—
316C4—61no—
317C4—121no—
318C4—121no—
319C4—242no—
320C4#540512yescomplemented
321C4D4×S411yescharacteristic
322C4Q8×S411yescharacteristic
323C3—41noSylow
324C2—61no—
325C2—61no—
326C2—122no—
327C2C4⋊C4×S411yescharacteristic
328C2C2×#540511yescharacteristic
329C2#8545211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2906D3
socleC22C2×C4×S4
prime radicalC2×#985C22
cosocleC2×#41⋊C3C23
hypercenterC4⋊C8S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. C4⋊(#131⋊C3)C2semidirect
4. #73122C2semidirect
5. C4⋊GU(2,𝔽3)C2semidirect
13. C4×GL(2,𝔽3)C4semidirect
14. C4×GL(2,𝔽3)C4semidirect
15. C4×#226C4semidirect
16. C4×#226C4semidirect
21. #2906D3semidirect
87. GL(2,𝔽3)C4⋊C4semidirect
88. #226C4⋊C4semidirect
320. C4#5405semidirect

Maximal subgroups

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

subgroupindex
C2×#54052
C4×#9852
C4⋊(#131⋊C3)2
#731222
C4⋊GU(2,𝔽3)2
#226863
#48374