Finite Group Databasebeta

← #11713926

Subgroups — #11713926

Order 768.

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

normal subgroups only

This group has 414 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⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#73065C211yesmaximal, characteristic
3#73065C211yesmaximal, characteristic
4#73065C211yesmaximal, characteristic
5#73063C211yesmaximal, characteristic, complemented
6C4⋊#859C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#27567—31nomaximal, Sylow
9C22×GL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#4994—41nomaximal
17#2512—31no—
18Q8×C4⋊C4D311yescharacteristic, complemented
19#2100—31no—
20#2099—31no—
21#2071—31no—
22#2065—31no—
23#1936—31no—
24#1919—31no—
25#1917—31no—
26#1880—31no—
27#1880—31no—
28#1880—31no—
29#1865—31no—
30#1803—31no—
31#1763—31no—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×GL(2,𝔽3)—21no—
34C2×GL(2,𝔽3)—21no—
35C2×GL(2,𝔽3)—21no—
36C2×C3×C4⋊C4—41no—
37C2×(C2×D3)⋊C4—41no—
38C2×(C2×D3)⋊C4—41no—
39C2×(C2×D3)⋊C4—41no—
40C2×C4⋊Dic3—41no—
41C2×Dic3⋊C4—41no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)—21no—
44C4×SL(2,𝔽3)—21no—
45#859—21no—
46#859—21no—
47#859—21no—
48C3⋊#95—41no—
49C22×SD16—31no—
50C2×#126—61no—
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53C2×C4×Q8C2×D311yescharacteristic
54C2×#106—31no—
55C2×C4⋊C8—31no—
56C2×#103—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#102—31no—
60C2×#102—31no—
61C2×#102—31no—
62C2×#97—31no—
63#397—61no—
64#396—61no—
65#390—31no—
66#390—31no—
67#390—31no—
68#389—31no—
69#389—31no—
70#389—31no—
71#388—61no—
72#384—31no—
73#384—31no—
74C4×C4⋊C4—31no—
75C4×C4⋊C4—31no—
76#340—31no—
77#340—31no—
78#340—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#336—31no—
85#336—31no—
86#336—31no—
87#336—31no—
88C23×D3—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C2×SL(2,𝔽3)#4111yescharacteristic
96C2×SL(2,𝔽3)#4111yescharacteristic
97C2×SL(2,𝔽3)#4111yescharacteristic
98GL(2,𝔽3)—41no—
99C3×C4⋊C4—82no—
100(C2×D3)⋊C4—82no—
101(C2×D3)⋊C4—82no—
102(C2×D3)⋊C4—82no—
103C4⋊Dic3—82no—
104Dic3⋊C4—82no—
105C22×Q8C22×D311yescharacteristic
106C22×D4—31no—
107C2×SD16—61no—
108C2×SD16—61no—
109C2×SD16—61no—
110C2×SD16—61no—
111C2×SD16—61no—
112C2×SD16—61no—
113C22×C8—31no—
114C22×C8—31no—
115C22×C8—31no—
116C22×C8—31no—
117#126—124no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—61no—
122C4×Q8—61no—
123C4×Q8—61no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4S411yescharacteristic
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C42—31no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—61no—
146#106—62no—
147C4⋊C8—121no—
148#103—61no—
149#103—61no—
150#103—61no—
151#103—61no—
152#103—61no—
153#103—61no—
154#102—62no—
155#102—121no—
156#102—121no—
157#97—121no—
158#95—31no—
159#95—31no—
160#95—61no—
161#95—61no—
162C23×C3—41no—
163C22×D3—42no—
164C22×D3—81no—
165C22×D3—81no—
166C22×D3—81no—
167C22×D3—81no—
168C22×D3—81no—
169C22×D3—81no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—81no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×Dic3—42no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188SL(2,𝔽3)#12611yescharacteristic, complemented
189C24—61no—
190C2×Q8—31no—
191C2×Q8—31no—
192C2×Q8—31no—
193C2×Q8#23511yescharacteristic
194C2×Q8#23711yescharacteristic
195C2×Q8#23911yescharacteristic
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—61no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—61no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4C2×S411yescharacteristic
210C22×C4C2×S411yescharacteristic
211C22×C4C2×S411yescharacteristic
212SD16—121no—
213SD16—121no—
214SD16—121no—
215SD16—121no—
216C2×C8—61no—
217C2×C8—61no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—61no—
228C4⋊C4—22no—
229C4⋊C4—32no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—122no—
243C4⋊C4—122no—
244C4⋊C4—241no—
245C4⋊C4—241no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C22⋊C4—122no—
250C22⋊C4—241no—
251C22⋊C4—241no—
252C42—61no—
253C42—61no—
254C42—61no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—122no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C2×D3—82no—
267C2×D3—82no—
268C2×D3—82no—
269C2×D3—161no—
270C2×D3—161no—
271C2×D3—161no—
272C2×D3—161no—
273C3×C4—82no—
274C3×C4—82no—
275C3×C4—161no—
276Dic3—82no—
277Dic3—161no—
278Dic3—161no—
279C23—121no—
280C23—121no—
281C23—121no—
282C23—121no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23C22×S411yescharacteristic
287Q8—31no—
288Q8—31no—
289Q8—31no—
290Q8—61no—
291Q8#89811yescharacteristic
292D4—121no—
293D4—121no—
294D4—121no—
295D4—121no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4#98312yes—
328C2×C4#98612yes—
329C8—121no—
330C8—121no—
331C8—121no—
332C8—121no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340D3—162no—
341C22—241no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22#591211yescharacteristic
349C22#591411yescharacteristic
350C22C2×#98311yescharacteristic
351C22#591911yescharacteristic
352C22C2×#98611yescharacteristic
353C22#592411yescharacteristic
354C22#592611yescharacteristic
355C4—22no—
356C4—22no—
357C4—31no—
358C4—31no—
359C4—31no—
360C4—31no—
361C4—41no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4—241no—
366C4—241no—
367C4—241no—
368C3—41noSylow
369C2—481no—
370C2#8531311yescharacteristic
371C2#8533211yescharacteristic
372C2#8536911yescharacteristic
373C2#8540111yescharacteristic
374C2#8541211yescharacteristic
375C2#8542211yescharacteristic
376C2#8543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C4⋊C4D3
prime radicalC22×GL(2,𝔽3)C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×SL(2,𝔽3)C2semidirect
5. #73063C2semidirect
6. C4⋊#859C2semidirect
7. #73059C2semidirect
18. Q8×C4⋊C4D3semidirect
188. SL(2,𝔽3)#126semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×SL(2,𝔽3)2
#730652
#730632
C4⋊#8592
#730592
#275673
#49944