Finite Group Databasebeta

← #11713024

Subgroups — #11713024

Order 768.

462 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 63 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×#5405C211yesmaximal, characteristic
2C2×#5404C211yesmaximal, characteristic
3#85341C211yesmaximal, characteristic
4C4⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
5#73122C211yesmaximal, characteristic, complemented
6#73122C211yesmaximal, characteristic, complemented
7C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
8#22687—31nomaximal, Sylow
9C2×#985C2211yescharacteristic
10C2×#131⋊C3C2211yescharacteristic
11C2×#131⋊C3C2211yescharacteristic
12C4×#41⋊C3C2211yescharacteristic
13#5424C2211yescharacteristic
14C2×GU(2,𝔽3)C2211yescharacteristic
15C2×GU(2,𝔽3)C2211yescharacteristic
16#5405—22no—
17#5404—22no—
18#5394C411yescharacteristic, complemented
19#5394C411yescharacteristic, complemented
20#5388C411yescharacteristic, complemented
21#5388C411yescharacteristic, complemented
22#4839—41nomaximal
23C2×#444—31no—
24C2×#443—31no—
25#2943—31no—
26#2906—31no—
27#2906D311yescharacteristic, complemented
28#1946—31no—
29#1945—31no—
30#1866—31no—
31#1866—31no—
32#1866—31no—
33#1866—31no—
34#1849—31no—
35#1806—31no—
36#1806—31no—
37#1771—31no—
38C2×#41⋊C3C2311yescharacteristic
39#985D412yes—
40#985Q812yes—
41C2×GL(2,𝔽3)C2×C411yescharacteristic
42C2×#226C2×C411yescharacteristic
43C2×ΓL(1,𝔽25)—41no—
44C2×D3×C8—41no—
45#872—41no—
46#131⋊C3—21no—
47#131⋊C3—21no—
48C4×SL(2,𝔽3)C2×C411yescharacteristic
49C4×SL(2,𝔽3)C2×C411yescharacteristic
50GU(2,𝔽3)—21no—
51GU(2,𝔽3)—21no—
52#859C2×C411yescharacteristic
53#859C2×C411yescharacteristic
54C3×C4⋊C8—41no—
55Dic3⋊C8—41no—
56Dic3⋊C8—41no—
57C4⋊(C3⋊C8)—41no—
58C2×#135—31no—
59C2×#131—31no—
60C2×#131—31no—
61C2×#131C2×D311yescharacteristic
62C2×#131C2×D311yescharacteristic
63C4×#41—31no—
64C4×#41C2×D311yescharacteristic
65#444—62no—
66#444—62no—
67#443—62no—
68#443—62no—
69#442—31no—
70#442—31no—
71#441—31no—
72#441—31no—
73#440—31no—
74#440—31no—
75#440—31no—
76#440—31no—
77C2×#108—31no—
78C2×#108—31no—
79C2×#108—31no—
80C2×#108—31no—
81#427—31no—
82#424—31no—
83#424—31no—
84#424—31no—
85#424—31no—
86#424—61no—
87#424—61no—
88#423—31no—
89#423—31no—
90#423—61no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8—61no—
93C2×#104—31no—
94C2×#104—31no—
95C2×#104—31no—
96C2×#104—31no—
97#416—31no—
98#416—31no—
99C2×#97—31no—
100C2×#97—31no—
101C2×C4×C8—31no—
102C4×C4⋊C4—31no—
103#339—31no—
104#339—31no—
105#339—31no—
106#339—31no—
107#336—31no—
108#336—31no—
109C2×C4×D3—41no—
110#41⋊C3C2×D411yescharacteristic
111#41⋊C3C2×Q811yescharacteristic
112C2×SL(2,𝔽3)C22×C411yescharacteristic
113GL(2,𝔽3)C4⋊C412yescomplemented
114#226C4⋊C412yescomplemented
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C42—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122C4×Dic3—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125ΓL(1,𝔽25)—82no—
126D3×C8—82no—
127C2×#41—31no—
128C2×#41C22×D311yescharacteristic
129#135—32no—
130#135—32no—
131#135—32no—
132#135—32no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×D8—31no—
137#131—21no—
138#131—21no—
139#131—61no—
140#131—61no—
141#131—62no—
142#131—62no—
143#131—121no—
144#131—121no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16—31no—
148C2×M16—31no—
149C2×M16—61no—
150C2×M16—61no—
151C22×C8—31no—
152C22×C8—31no—
153C22×C8—31no—
154C22×C8—31no—
155C22×C8—61no—
156C22×C8—61no—
157C4×Q8—31no—
158C4×Q8—31no—
159C4×Q8C4×D311yescharacteristic
160C4×Q8C4×D311yescharacteristic
161C4×D4—31no—
162C4×D4—31no—
163C4×D4—31no—
164C4×D4—31no—
165C4×D4—61no—
166C4×D4—61no—
167#117—31no—
168#117—31no—
169#117—61no—
170C2×C4⋊C4—61no—
171C2×C42—31no—
172C2×C42—31no—
173C2×C42—31no—
174C2×C42—61no—
175#108—62no—
176#108—62no—
177#108—121no—
178#108—121no—
179#107—31no—
180#107—31no—
181#106—31no—
182#106—31no—
183C4⋊C8—31no—
184C4⋊C8—31no—
185C4⋊C8—31no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8S411yescharacteristic
194#104—62no—
195#104—61no—
196#104—62no—
197#104—61no—
198#104—61no—
199#104—61no—
200#103—31no—
201#103—31no—
202#103—31no—
203#103—31no—
204#102—31no—
205#102—31no—
206#102—31no—
207#102—31no—
208#97—31no—
209#97—31no—
210#97—31no—
211#97—31no—
212#97—62no—
213C4×C8—62no—
214#95—61no—
215C22×D3—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C4×D3—42no—
223C4×D3—42no—
224SL(2,𝔽3)C2×C4⋊C411yescharacteristic
225C3×C8—81no—
226C3×C8—81no—
227C3⋊C8—81no—
228C3⋊C8—81no—
229#41—31no—
230#41—31no—
231#41—32no—
232#41—32no—
233#41—61no—
234#41—61no—
235#41D3×D411yescharacteristic
236#41D3×Q811yescharacteristic
237C2×Q8—31no—
238C2×Q8C2×C4×D311yescharacteristic
239C2×D4—31no—
240C2×D4—31no—
241C2×D4—61no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250Q16—32no—
251Q16—32no—
252SD16—32no—
253SD16—32no—
254SD16—32no—
255SD16—32no—
256D8—32no—
257D8—32no—
258M16—61no—
259M16—61no—
260M16—61no—
261M16—61no—
262M16—61no—
263M16—61no—
264M16—61no—
265M16—61no—
266M16—122no—
267M16—122no—
268C2×C8—32no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—32no—
272C2×C8—31no—
273C2×C8—31no—
274C2×C8—31no—
275C2×C8—31no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—122no—
285C2×C8—122no—
286C2×C8C2×S411yescharacteristic
287C2×C8C2×S411yescharacteristic
288C4⋊C4—31no—
289C4⋊C4—31no—
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—122no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C42—31no—
300C42—31no—
301C42—31no—
302C42—61no—
303C42—61no—
304C42—61no—
305C42—61no—
306C42—61no—
307C42—61no—
308C42—61no—
309C42C2×S411yescharacteristic
310C22×C3—41no—
311C2×D3—42no—
312C2×D3—81no—
313C2×D3—81no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—81no—
317C3×C4—81no—
318Dic3—42no—
319Dic3—81no—
320Dic3—81no—
321C23—31no—
322C23—61no—
323Q8—31no—
324Q8—32no—
325Q8—61no—
326Q8D3×C4⋊C411yescharacteristic
327D4—31no—
328D4—31no—
329D4—32no—
330D4—61no—
331D4—61no—
332D4—62no—
333D4—62no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—61no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—241no—
356C2×C4C4×S411yescharacteristic
357C2×C4C4×S411yescharacteristic
358C2×C4C22×S411yescharacteristic
359C8—21no—
360C8—21no—
361C8—32no—
362C8—32no—
363C8—61no—
364C8—61no—
365C8—61no—
366C8—61no—
367C8—121no—
368C8—121no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—41no—
372D3—82no—
373C22—31no—
374C22—31no—
375C22—61no—
376C22—61no—
377C22—62no—
378C22—121no—
379C22—121no—
380C22C2×C4×S411yescharacteristic
381C4—21no—
382C4—21no—
383C4—31no—
384C4—31no—
385C4—62no—
386C4—61no—
387C4—61no—
388C4—122no—
389C4—121no—
390C4—121no—
391C4—241no—
392C4D4×S411yescharacteristic
393C4Q8×S411yescharacteristic
394C3—41noSylow
395C2—61no—
396C2—61no—
397C2—122no—
398C2C4⋊C4×S411yescharacteristic
399C2#8544811yescharacteristic
400C2#8545311yescharacteristic
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
4. C4⋊(#131⋊C3)C2semidirect
5. #73122C2semidirect
6. #73122C2semidirect
7. C4⋊GU(2,𝔽3)C2semidirect
18. #5394C4semidirect
19. #5394C4semidirect
20. #5388C4semidirect
21. #5388C4semidirect
27. #2906D3semidirect
113. GL(2,𝔽3)C4⋊C4semidirect
114. #226C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#54052
C2×#54042
#853412
C4⋊(#131⋊C3)2
#731222
C4⋊GU(2,𝔽3)2
#226873
#48394