Finite Group Databasebeta

← #11713969

Subgroups — #11713969

Order 768.

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

normal subgroups only

This group has 719 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, complemented
2C2×#5427C212yesmaximal, complemented
3C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C212yesmaximal, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#32904—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×GL(2,𝔽3)C2212yes—
9C4×#41⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12#5427—24no—
13C2×#859C2212yes—
14C4⋊GL(2,𝔽3)—22no—
15#4939—41nomaximal
16C2×C4×#41D311yescharacteristic, complemented
17C2×#460—31no—
18C2×#459—31no—
19C2×#449—32no—
20#2856—31no—
21#1880—32no—
22#1879—32no—
23#1875—32no—
24#1803—31no—
25#1802—31no—
26#1797—31no—
27C22×D12—41no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3—21no—
30C2×#41⋊C3D412yescomplemented
31C22×SL(2,𝔽3)C2311yescharacteristic
32C2×GL(2,𝔽3)—22no—
33C2×GL(2,𝔽3)—22no—
34C2×GL(2,𝔽3)—22no—
35C2×C3×C42—41no—
36C2×(C2×D3)⋊C4—42no—
37C2×(C2×D3)⋊C4—42no—
38C2×C4⋊Dic3—41no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)D412yescomplemented
41#859—22no—
42C22×#41C2×D311yescharacteristic
43C22×SD16—32no—
44C22×D8—32no—
45C2×#127—31no—
46C2×#121—32no—
47C2×#121—61no—
48C2×#121—61no—
49C4×#41—22no—
50C4×#41—62no—
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53C2×C4×D4—31no—
54C2×C4×D4—31no—
55C2×#117—31no—
56C22×C42—31no—
57#460—62no—
58#460—62no—
59#459—64no—
60#449—64no—
61#449—64no—
62C2×C4⋊C8—31no—
63C2×C4⋊C8—31no—
64C2×#103—32no—
65C2×#102—32no—
66C2×#102—32no—
67C2×#102—32no—
68C2×C22⋊C8—32no—
69#390—32no—
70#390—61no—
71#390—61no—
72#384—31no—
73#340—32no—
74#340—32no—
75#336—32no—
76C23×D3—42no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×Dic3—42no—
81C2×D12—44no—
82C2×D12—82no—
83C2×D12—82no—
84#41⋊C3—22no—
85C2×SL(2,𝔽3)C2×D411yescharacteristic
86C2×SL(2,𝔽3)C2×D411yescharacteristic
87C2×SL(2,𝔽3)#4111yescharacteristic
88GL(2,𝔽3)—42no—
89C3×C42—82no—
90(C2×D3)⋊C4—84no—
91(C2×D3)⋊C4—84no—
92C4⋊Dic3—82no—
93C2×#41—21no—
94C2×#41—21no—
95C2×#41—32no—
96C2×#41—61no—
97C2×#41—61no—
98C2×#41D1212yescomplemented
99C22×Q8C22×D311yescharacteristic
100C22×D4—31no—
101C22×D4—32no—
102C22×D4—61no—
103C22×D4—62no—
104C22×D4—61no—
105C23×C4—31no—
106C23×C4—31no—
107C23×C4—31no—
108C2×SD16—62no—
109C2×SD16—62no—
110C2×SD16—62no—
111C2×SD16—62no—
112C2×SD16—62no—
113C2×SD16—62no—
114C2×D8—64no—
115C2×D8—64no—
116C2×D8—64no—
117C22×C8—32no—
118C22×C8—32no—
119#127—62no—
120#127—62no—
121#121—64no—
122#121—64no—
123#121—124no—
124#121—124no—
125C4×Q8—21no—
126C4×Q8—32no—
127C4×Q8—61no—
128C4×Q8#6712yes—
129C4×D4—34no—
130C4×D4—62no—
131C4×D4—62no—
132C4×D4—62no—
133#117—62no—
134#117—62no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—32no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—62no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147C2×C42—62no—
148C2×C42—62no—
149C2×C42S411yescharacteristic
150C4⋊C8—62no—
151C4⋊C8—62no—
152#103—62no—
153#103—62no—
154#102—64no—
155#102—64no—
156#102—64no—
157C22⋊C8—64no—
158C23×C3—41no—
159C22×D3—44no—
160C22×D3—82no—
161C22×D3—82no—
162C22×D3—82no—
163C22×D3—82no—
164C22×D3—82no—
165C22×D3—82no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×Dic3—82no—
176C2×Dic3—82no—
177C2×Dic3—82no—
178D12—88no—
179SL(2,𝔽3)#12111yescharacteristic, complemented
180C24—31no—
181C24—62no—
182#41—22no—
183#41—64no—
184#41—62no—
185#41—62no—
186#41—62no—
187C2×Q8—31no—
188C2×Q8—31no—
189C2×Q8—31no—
190C2×Q8C2×D1211yescharacteristic
191C2×Q8#23511yescharacteristic
192C2×Q8C2×#6711yescharacteristic
193C2×D4—32no—
194C2×D4—32no—
195C2×D4—32no—
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—61no—
199C2×D4—62no—
200C2×D4—62no—
201C2×D4—62no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—62no—
205C2×D4—122no—
206C2×D4—122no—
207C2×D4—122no—
208C2×D4—124no—
209C2×D4—124no—
210C2×D4—124no—
211C2×D4—122no—
212C2×D4—122no—
213C2×D4—122no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—32no—
217C22×C4—32no—
218C22×C4—31no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—62no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—121no—
230C22×C4C2×S411yescharacteristic
231C22×C4C2×S411yescharacteristic
232C22×C4C2×S411yescharacteristic
233SD16—122no—
234SD16—122no—
235SD16—122no—
236SD16—122no—
237D8—128no—
238C2×C8—62no—
239C2×C8—62no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C4⋊C4—32no—
245C4⋊C4—32no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—122no—
251C4⋊C4—122no—
252C22⋊C4—64no—
253C22⋊C4—124no—
254C22⋊C4—124no—
255C22⋊C4—124no—
256C42—22no—
257C42—32no—
258C42—32no—
259C42—62no—
260C42—61no—
261C42—61no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C2×D3—84no—
270C2×D3—84no—
271C2×D3—84no—
272C2×D3—162no—
273C2×D3—162no—
274C2×D3—162no—
275C2×D3—162no—
276C3×C4—44no—
277C3×C4—82no—
278C3×C4—82no—
279Dic3—162no—
280C23—32no—
281C23—61no—
282C23—61no—
283C23—61no—
284C23—61no—
285C23—61no—
286C23—61no—
287C23—122no—
288C23—122no—
289C23—122no—
290C23—122no—
291C23—122no—
292C23—122no—
293C23—122no—
294C23C22×S411yescharacteristic
295Q8—31no—
296Q8—31no—
297Q8—31no—
298Q8—61no—
299Q8#93011yescharacteristic
300D4—34no—
301D4—62no—
302D4—62no—
303D4—62no—
304D4—122no—
305D4—122no—
306D4—122no—
307D4—122no—
308D4—244no—
309D4—248no—
310D4—244no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—32no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—124no—
332C2×C4—124no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4C4⋊S412yes—
337C2×C4#98512yes—
338C2×C4#98612yes—
339C2×C4#98612yes—
340C2×C4GL(2,ℤ/4)12yes—
341C8—122no—
342C8—122no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350D3—164no—
351C22—62no—
352C22—62no—
353C22—62no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—242no—
359C22—242no—
360C22—242no—
361C22—242no—
362C22—242no—
363C22—242no—
364C22—242no—
365C22C2×C4⋊S411yescharacteristic
366C22#591211yescharacteristic
367C22C2×#98511yescharacteristic
368C22C2×#98611yescharacteristic
369C22C2×#98611yescharacteristic
370C22#592311yescharacteristic
371C22C2×GL(2,ℤ/4)11yescharacteristic
372C4—22no—
373C4—22no—
374C4—31no—
375C4—31no—
376C4—31no—
377C4—31no—
378C4—62no—
379C4—121no—
380C4—242no—
381C4#542714yescomplemented
382C3—41noSylow
383C2—122no—
384C2—482no—
385C2#8534411yescharacteristic
386C2#8534511yescharacteristic
387C2#8534611yescharacteristic
388C2#8534811yescharacteristic
389C2#8548611yescharacteristic
390C2C2×#542711yescharacteristic
391C2#8552311yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×C4×#41D3
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
1. C2×C4×#41⋊C3C2semidirect
2. C2×#5427C2semidirect
3. C2×C4⋊GL(2,𝔽3)C2semidirect
4. #73065C2semidirect
5. C4⋊#859C2semidirect
16. C2×C4×#41D3semidirect
30. C2×#41⋊C3D4semidirect
40. C4×SL(2,𝔽3)D4semidirect
98. C2×#41D12semidirect
179. SL(2,𝔽3)#121semidirect
381. C4#5427semidirect

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×#54272
C2×C4⋊GL(2,𝔽3)2
#730652
C4⋊#8592
#329043
#49394