Finite Group Databasebeta

← #11713804

Subgroups — #11713804

Order 768.

441 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 61 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⋊#226C211yesmaximal, characteristic
2C2×C4×#226C211yesmaximal, characteristic
3C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
4#73062C211yesmaximal, characteristic
5#73061C211yesmaximal, characteristic
6#73061C211yesmaximal, characteristic
7#73061C211yesmaximal, characteristic
8#26171—31nomaximal, Sylow
9C22×#226C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C4⋊#226—22no—
17C4×#226—22no—
18#4980—41nomaximal
19C2×#462—31no—
20C2×C4×Q16—31no—
21#2446—31no—
22Q8×C4⋊C4D311yescharacteristic
23#2095—31no—
24#2087—31no—
25#2072—31no—
26#2043—31no—
27#1933—31no—
28#1916—31no—
29#1878—31no—
30#1878—31no—
31#1878—31no—
32#1863—31no—
33#1863—31no—
34C22×SL(2,𝔽3)C2311yescharacteristic
35C2×#226—21no—
36C2×#226—21no—
37C2×#226—21no—
38C2×#226—21no—
39C2×#226—21no—
40C2×#226D412yes—
41C2×C3×C4⋊C4—41no—
42C2×C4⋊Dic3—41no—
43C2×C4×Dic3—41no—
44C4×SL(2,𝔽3)—21no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)—21no—
47#859—21no—
48#859—21no—
49C3⋊#95—41no—
50C3⋊#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C22×Q16—31no—
54C22×Q16—31no—
55C2×C4⋊Q8—31no—
56C2×C4×Q8—31no—
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60#462—62no—
61#462—62no—
62C4×Q16—62no—
63C4×Q16—62no—
64C2×#107—31no—
65C2×#107—31no—
66C2×C4⋊C8—31no—
67C2×#103—31no—
68C2×#103—31no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×C4×C8—31no—
76#400—61no—
77#398—61no—
78#391—31no—
79#391—31no—
80#391—31no—
81#389—31no—
82#389—31no—
83#389—31no—
84#387—61no—
85#387—61no—
86#384—31no—
87C4×C4⋊C4—31no—
88#340—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#336—31no—
93#336—31no—
94#336—31no—
95#336—31no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C2×SL(2,𝔽3)C2×D411yescharacteristic
104C2×SL(2,𝔽3)#4111yescharacteristic
105C2×SL(2,𝔽3)#4111yescharacteristic
106#226—22no—
107#226—41no—
108C3×C4⋊C4—44no—
109C4⋊Dic3—82no—
110C4×Dic3—82no—
111C22×Q8—31no—
112C22×Q8—31no—
113C22×Q8C22×D311yescharacteristic
114C2×Q16—32no—
115C2×Q16—32no—
116C2×Q16—61no—
117C2×Q16—61no—
118C2×Q16—61no—
119C2×Q16—61no—
120C2×Q16—61no—
121C2×Q16—61no—
122C2×Q16—61no—
123C2×Q16—61no—
124C2×Q16—61no—
125C2×Q16—61no—
126C22×C8—31no—
127C22×C8—31no—
128C22×C8—31no—
129C22×C8—31no—
130C4⋊Q8—62no—
131C4⋊Q8—121no—
132C4×Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—61no—
136C4×Q8—61no—
137C4×Q8—61no—
138C4×Q8—62no—
139C4×Q8—121no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4S411yescharacteristic
152C2×C42—31no—
153C2×C42—31no—
154C2×C42—31no—
155C2×C42—61no—
156#107—62no—
157#107—121no—
158C4⋊C8—62no—
159#103—62no—
160#103—62no—
161#103—62no—
162#103—61no—
163#103—62no—
164#103—61no—
165#103—62no—
166#103—61no—
167#103—61no—
168#103—121no—
169C4×C8—62no—
170#95—31no—
171#95—31no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177#95—61no—
178#95—61no—
179C23×C3—41no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201SL(2,𝔽3)#12311yescharacteristic
202C2×Q8—31no—
203C2×Q8—31no—
204C2×Q8—31no—
205C2×Q8—32no—
206C2×Q8—61no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8#23511yescharacteristic
218C2×Q8D3×D411yescharacteristic
219C2×Q8#23911yescharacteristic
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C2×S411yescharacteristic
229C22×C4C2×S411yescharacteristic
230C22×C4C2×S411yescharacteristic
231Q16—62no—
232Q16—62no—
233Q16—62no—
234Q16—62no—
235Q16—121no—
236Q16—121no—
237Q16—121no—
238Q16—121no—
239C2×C8—32no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—61no—
243C2×C8—61no—
244C2×C8—61no—
245C2×C8—61no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—61no—
251C4⋊C4—32no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—121no—
262C4⋊C4—122no—
263C4⋊C4—122no—
264C4⋊C4—122no—
265C4⋊C4—122no—
266C4⋊C4—241no—
267C4⋊C4#22614yescomplemented
268C42—61no—
269C42—61no—
270C42—61no—
271C42—61no—
272C42—61no—
273C42—61no—
274C42—122no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C3×C4—82no—
283C3×C4—82no—
284C3×C4—82no—
285Dic3—82no—
286Dic3—82no—
287Dic3—161no—
288Dic3—161no—
289C23C22×S411yescharacteristic
290Q8—31no—
291Q8—31no—
292Q8—31no—
293Q8—61no—
294Q8—62no—
295Q8—62no—
296Q8—121no—
297Q8—121no—
298Q8—121no—
299Q8—121no—
300Q8—121no—
301Q8—121no—
302Q8#89411yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4C2×#22612yes—
337C2×C4C2×#22612yes—
338C2×C4C2×#22612yes—
339C8—62no—
340C8—121no—
341C8—121no—
342C8—121no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C22#591211yescharacteristic
351C22D4×S411yescharacteristic
352C22C22×#22611yescharacteristic
353C22#591911yescharacteristic
354C22#592311yescharacteristic
355C22#592411yescharacteristic
356C22#592711yescharacteristic
357C4—22no—
358C4—22no—
359C4—22no—
360C4—31no—
361C4—31no—
362C4—31no—
363C4—31no—
364C4—122no—
365C4—121no—
366C4—121no—
367C4—121no—
368C4—241no—
369C4—241no—
370C4—241no—
371C3—41noSylow
372C2#8530911yescharacteristic
373C2#8532211yescharacteristic
374C2#8536211yescharacteristic
375C2D4×#22611yescharacteristic
376C2#8541111yescharacteristic
377C2#8542511yescharacteristic
378C2#8542811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C4⋊C4D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
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
267. C4⋊C4#226semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#2262
C2×C4×#2262
C4⋊C4×SL(2,𝔽3)2
#730622
#730612
#261713
#49804