Finite Group Databasebeta

← #11714031

Subgroups — #11714031

Order 768.

457 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C4×2−1+4⋊C3C211yesmaximal, characteristic
2#85339C211yesmaximal, characteristic, complemented
3#73155C211yesmaximal, characteristic
4#73155C211yesmaximal, characteristic
5#73076C211yesmaximal, characteristic, complemented
6#73075C211yesmaximal, characteristic, complemented
7C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
8#32974—31nomaximal, Sylow
9C2×2−1+4⋊C3C2211yescharacteristic
10#520⋊C3C2211yescharacteristic
11#519⋊C3—21no—
12C4×#41⋊C3C2211yescharacteristic
13C2×C4×SL(2,𝔽3)—21no—
14#5426C2211yescharacteristic
15#5426C2211yescharacteristic
16C2×GU(2,𝔽3)C2211yescharacteristic
17C2×GU(2,𝔽3)C2211yescharacteristic
18#5389—21no—
19#5389—21no—
20#5009—41nomaximal
21C4×2−1+4D311yescharacteristic
22#3080—31no—
23#3080—31no—
24#2565—31no—
25#1860—31no—
26#1860—31no—
27#1859—31no—
28#1859—31no—
29#1857—31no—
30#1857—31no—
31#1806—31no—
32#1806—31no—
33#1801—31no—
34#1793—31no—
35#1793—31no—
362−1+4⋊C3—21no—
372−1+4⋊C3Q812yes—
38C2×#41⋊C3C2311yescharacteristic
39C22×SL(2,𝔽3)—21no—
40C3×C4×D4—41no—
41#870—41no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)D411yescharacteristic
44C4×SL(2,𝔽3)D411yescharacteristic
45GU(2,𝔽3)—21no—
46GU(2,𝔽3)—21no—
47#859—21no—
48#859—21no—
49#832—41no—
50#832—41no—
51#808—41no—
52#807—41no—
53C4⋊(C3⋊C8)—41no—
54C2×2−1+4C2×D311yescharacteristic
55C2×#125—31no—
56#520—31no—
57#520—61no—
58#520C2×D311yescharacteristic
59#519—21no—
60#519—31no—
61C4×#41—31no—
62C4×#41—61no—
63C4×#41C2×D311yescharacteristic
64C2×C4×Q8—21no—
65C2×C4×Q8—31no—
66#479—61no—
67#479—61no—
68#479—61no—
69#479—61no—
70#478—61no—
71#478—61no—
72#478—61no—
73#478—61no—
74#428—31no—
75#428—31no—
76#428—31no—
77#428—31no—
78#423—31no—
79#423—31no—
80C2×#104—31no—
81C2×#104—31no—
82C2×#104—31no—
83C2×#104—31no—
84#417—31no—
85#417—31no—
86#417—31no—
87#417—31no—
88C2×#101—31no—
89C2×#101—31no—
90#398—62no—
91#391—31no—
92#391—31no—
93#387—62no—
94#384—31no—
95#382—31no—
96C4×C4⋊C4—31no—
97#343—31no—
98#343—31no—
99#339—31no—
100#339—31no—
101#339—31no—
102#339—31no—
103C2×C3×D4—41no—
104C22×C3×C4—81no—
105#41⋊C3C2×Q811yescharacteristic
106#41⋊C3#4111yescharacteristic
107C2×SL(2,𝔽3)—22no—
108C2×SL(2,𝔽3)C2×D411yescharacteristic
109C3×C4⋊C4—41no—
110C3×C22⋊C4—81no—
111C3×C42—41no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114Dic3⋊C4—81no—
115Dic3⋊C4—81no—
116C2×C3⋊C8—41no—
117C2×C3⋊C8—41no—
1182−1+4—21no—
1192−1+4—32no—
1202−1+4—61no—
1212−1+4Dic612yes—
122C2×#41—31no—
123C2×#41—61no—
124C2×#41C22×D311yescharacteristic
125C22×Q8—21no—
126C22×Q8—31no—
127C2×M16—31no—
128C2×M16—31no—
129C2×M16—31no—
130C2×M16—31no—
131#125—61no—
132#125—61no—
133#125—61no—
134#125—61no—
135C4×Q8—21no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8—61no—
141C4×Q8—61no—
142C4×Q8—61no—
143C4×Q8#6711yescharacteristic
144C4×Q8#6711yescharacteristic
145C4×D4—31no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4—62no—
151C4×D4—61no—
152C4×D4S411yescharacteristic
153#117—31no—
154#117—31no—
155#117—61no—
156#117—62no—
157#117—61no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—32no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—62no—
167C2×C42—31no—
168C2×C42—32no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42—61no—
172#107—61no—
173#107—61no—
174#107—61no—
175#107—61no—
176#106—61no—
177#106—61no—
178#106—61no—
179#106—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184#104—61no—
185#104—61no—
186#104—62no—
187#104—62no—
188#104—61no—
189#104—61no—
190#103—61no—
191#103—61no—
192#103—61no—
193#103—61no—
194#102—61no—
195#102—61no—
196#102—61no—
197#102—61no—
198#101—62no—
199#101—62no—
200#95—61no—
201#95—62no—
202#95—62no—
203#95—62no—
204C23×C3—81no—
205C3×D4—42no—
206C3×D4—81no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—81no—
211C2×C3×C4—162no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214SL(2,𝔽3)C22⋊Q811yescharacteristic
215C3⋊C8—81no—
216C3⋊C8—81no—
217#41—31no—
218#41—31no—
219#41—32no—
220#41—32no—
221#41—62no—
222#41—61no—
223#41—61no—
224#41—64no—
225#41—62no—
226#41C2×Dic611yescharacteristic
227#41#23511yescharacteristic
228C2×Q8—22no—
229C2×Q8—32no—
230C2×Q8—32no—
231C2×Q8—31no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—62no—
236C2×Q8C2×#6711yescharacteristic
237C2×D4—31no—
238C2×D4—61no—
239C2×D4C2×S411yescharacteristic
240C22×C4—21no—
241C22×C4—31no—
242C22×C4—32no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—32no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252M16—61no—
253M16—61no—
254M16—61no—
255M16—61no—
256M16—61no—
257M16—61no—
258M16—61no—
259M16—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C4⋊C4—31no—
265C4⋊C4—31no—
266C4⋊C4—31no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—122no—
278C4⋊C4—124no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4C2×S411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C42—31no—
287C42—31no—
288C42—31no—
289C42—61no—
290C42—61no—
291C42—62no—
292C42—61no—
293C42—61no—
294C42—61no—
295C42—61no—
296C42—61no—
297C42C2×S411yescharacteristic
298C22×C3—41no—
299C22×C3—82no—
300C22×C3—161no—
301C22×C3—161no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4—161no—
307Dic3—161no—
308Dic3—161no—
309C23—21no—
310C23—31no—
311Q8—31no—
312Q8—32no—
313Q8—62no—
314Q8—62no—
315Q8—61no—
316Q8#92411yescharacteristic
317D4—21no—
318D4—31no—
319D4—31no—
320D4—61no—
321D4—62no—
322D4—62no—
323D4A4⋊Q812yes—
324C2×C4—21no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—34no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—42no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—61no—
338C2×C4—62no—
339C2×C4—61no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4GL(2,ℤ/4)11yescharacteristic
354C2×C4GL(2,ℤ/4)11yescharacteristic
355C2×C4C22×S411yescharacteristic
356C8—121no—
357C8—121no—
358C8—121no—
359C8—121no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—162no—
364C22—22no—
365C22—31no—
366C22—31no—
367C22—41no—
368C22—41no—
369C22—61no—
370C22—61no—
371C22C2×GL(2,ℤ/4)11yescharacteristic
372C4—21no—
373C4—21no—
374C4—31no—
375C4—31no—
376C4—41no—
377C4—61no—
378C4—61no—
379C4—62no—
380C4—122no—
381C4—121no—
382C4—241no—
383C4—241no—
384C4—241no—
385C4C2×A4⋊Q811yescharacteristic
386C4#591211yescharacteristic
387C3—41noSylow
388C2—42no—
389C2—61no—
390C2—61no—
391C2#8548011yescharacteristic
392C2#8553311yescharacteristic
393C2#8554411yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×2−1+4D3
prime radicalC2×2−1+4⋊C3C22
hypercenterC4×D4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #85339C2semidirect
5. #73076C2semidirect
6. #73075C2semidirect
7. C4⋊GU(2,𝔽3)C2semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4⋊C32
#853392
#731552
#730762
#730752
C4⋊GU(2,𝔽3)2
#329743
#50094