Finite Group Databasebeta

← #11714030

Subgroups — #11714030

Order 768.

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

normal subgroups only

This group has 469 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×#994C211yesmaximal, characteristic
2#85339C211yesmaximal, characteristic, complemented
3#73153C211yesmaximal, characteristic
4#73153C211yesmaximal, characteristic
5#73076C211yesmaximal, characteristic, complemented
6#73075C211yesmaximal, characteristic, complemented
7C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
8#32973—31nomaximal, Sylow
9C2×#994C2211yescharacteristic
10#520⋊C3—21no—
11#520⋊C3C2211yescharacteristic
12C4×#41⋊C3—21no—
13C4×#41⋊C3C2211yescharacteristic
14#5426C2211yescharacteristic
15#5426C2211yescharacteristic
16C2×GU(2,𝔽3)C2211yescharacteristic
17C2×GU(2,𝔽3)C2211yescharacteristic
18#5389—21no—
19#5389—21no—
20#5022—41nomaximal
21C4×H(5,𝔽2)D311yescharacteristic
22#3080—31no—
23#3080—31no—
24#2566—31no—
25#1860—31no—
26#1860—31no—
27#1859—31no—
28#1859—31no—
29#1856—31no—
30#1856—31no—
31#1806—31no—
32#1806—31no—
33#1800—31no—
34#1792—31no—
35#1792—31no—
36#994—21no—
37#994Q812yes—
38C2×#41⋊C3—21no—
39C2×#41⋊C3C2311yescharacteristic
40C3×C4×Q8—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#835—41no—
50#835—41no—
51#808—41no—
52#807—41no—
53C4⋊(C3⋊C8)—41no—
54C2×H(5,𝔽2)C2×D311yescharacteristic
55C2×#125—31no—
56#520—21no—
57#520—31no—
58#520C2×D311yescharacteristic
59#518—31no—
60#518—61no—
61C4×#41—21no—
62C4×#41—31no—
63C4×#41C2×D311yescharacteristic
64C2×C4×D4—31no—
65C2×C4×D4—61no—
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×#100—31no—
89C2×#100—31no—
90#396—62no—
91#390—31no—
92#390—31no—
93#385—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×Q8—41no—
104#41⋊C3—22no—
105#41⋊C3C2×Q811yescharacteristic
106#41⋊C3#4111yescharacteristic
107C2×SL(2,𝔽3)C2×D411yescharacteristic
108C3×C4⋊C4—41no—
109C3×C4⋊C4—81no—
110C3×C42—41no—
111C3×C42—81no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114Dic3⋊C4—81no—
115Dic3⋊C4—81no—
116C2×C3⋊C8—41no—
117C2×C3⋊C8—41no—
118H(5,𝔽2)—21no—
119H(5,𝔽2)—32no—
120H(5,𝔽2)—61no—
121H(5,𝔽2)Dic612yes—
122C2×#41—21no—
123C2×#41—31no—
124C2×#41C22×D311yescharacteristic
125C22×D4—31no—
126C22×D4—61no—
127C23×C4—32no—
128C2×M16—31no—
129C2×M16—31no—
130C2×M16—31no—
131C2×M16—31no—
132#125—61no—
133#125—61no—
134#125—61no—
135#125—61no—
136C4×Q8—21no—
137C4×Q8—31no—
138C4×Q8#6711yescharacteristic
139C4×Q8#6711yescharacteristic
140C4×Q8S411yescharacteristic
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—31no—
144C4×D4—31no—
145C4×D4—61no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4—61no—
151C4×D4—62no—
152C4×D4—61no—
153C4×D4—61no—
154#117—31no—
155#117—31no—
156#117—61no—
157#117—61no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—62no—
166C2×C22⋊C4—32no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C42—31no—
171C2×C42—31no—
172C2×C42—31no—
173C2×C42—61no—
174#107—61no—
175#107—61no—
176#107—61no—
177#107—61no—
178#106—61no—
179#106—61no—
180#106—61no—
181#106—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186#104—61no—
187#104—61no—
188#104—62no—
189#104—62no—
190#104—61no—
191#104—61no—
192#103—61no—
193#103—61no—
194#103—61no—
195#103—61no—
196#102—61no—
197#102—61no—
198#102—61no—
199#102—61no—
200#100—62no—
201#100—62no—
202#95—61no—
203#95—62no—
204C3×Q8—42no—
205C3×Q8—81no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213SL(2,𝔽3)C22⋊Q811yescharacteristic
214C3⋊C8—81no—
215C3⋊C8—81no—
216C24—32no—
217#41—22no—
218#41—31no—
219#41—31no—
220#41—32no—
221#41—32no—
222#41—61no—
223#41—61no—
224#41—62no—
225#41C2×Dic611yescharacteristic
226#41#23511yescharacteristic
227C2×Q8C2×#6711yescharacteristic
228C2×Q8C2×S411yescharacteristic
229C2×D4—31no—
230C2×D4—32no—
231C2×D4—32no—
232C2×D4—31no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—64no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—32no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—62no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—242no—
253M16—61no—
254M16—61no—
255M16—61no—
256M16—61no—
257M16—61no—
258M16—61no—
259M16—61no—
260M16—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C4⋊C4—21no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—31no—
269C4⋊C4—61no—
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—121no—
279C4⋊C4—121no—
280C4⋊C4—242no—
281C4⋊C4C2×S411yescharacteristic
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—62no—
286C22⋊C4—61no—
287C22⋊C4—124no—
288C22⋊C4—242no—
289C42—21no—
290C42—31no—
291C42—31no—
292C42—31no—
293C42—61no—
294C42—61no—
295C42—61no—
296C42—61no—
297C42—61no—
298C42—61no—
299C42C2×S411yescharacteristic
300C22×C3—41no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4—82no—
306C3×C4—161no—
307Dic3—161no—
308Dic3—161no—
309C23—31no—
310C23—34no—
311C23—62no—
312C23—61no—
313C23—122no—
314C23—122no—
315Q8—21no—
316Q8—31no—
317Q8#92411yescharacteristic
318Q8A4⋊Q812yes—
319D4—31no—
320D4—31no—
321D4—32no—
322D4—62no—
323D4—62no—
324D4—61no—
325D4—62no—
326D4—62no—
327D4—61no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—241no—
355C2×C4—242no—
356C2×C4GL(2,ℤ/4)11yescharacteristic
357C2×C4GL(2,ℤ/4)11yescharacteristic
358C2×C4C22×S411yescharacteristic
359C8—121no—
360C8—121no—
361C8—121no—
362C8—121no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—41no—
366C22—31no—
367C22—31no—
368C22—61no—
369C22—61no—
370C22—62no—
371C22—124no—
372C22—121no—
373C22—121no—
374C22—242no—
375C22C2×GL(2,ℤ/4)11yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—31no—
380C4—31no—
381C4—41no—
382C4—61no—
383C4—61no—
384C4—122no—
385C4—121no—
386C4—241no—
387C4—241no—
388C4—241no—
389C4C2×A4⋊Q811yescharacteristic
390C4#591211yescharacteristic
391C3—41noSylow
392C2—61no—
393C2—61no—
394C2—122no—
395C2#8548011yescharacteristic
396C2#8552611yescharacteristic
397C2#8554411yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×H(5,𝔽2)D3
prime radicalC2×#994C22
hypercenterC4×Q8S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

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×#9942
#853392
#731532
#730762
#730752
C4⋊GU(2,𝔽3)2
#329733
#50224