Finite Group Databasebeta

← #11714024

Subgroups — #11714024

Order 768.

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

normal subgroups only

This group has 512 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×#519⋊C3C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic, complemented
3#73148C211yesmaximal, characteristic
4#73148C211yesmaximal, characteristic
5#73062C211yesmaximal, characteristic, complemented
6C4⋊#859C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#32967—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10#519⋊C3—22no—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5389—22no—
18#4948—41nomaximal
19C2×#519D311yescharacteristic
20C2×#479—31no—
21C2×#479—31no—
22#2629—31no—
23#1865—31no—
24#1865—31no—
25#1863—31no—
26#1863—31no—
27#1854—31no—
28#1854—31no—
29#1803—31no—
30#1803—31no—
31#1798—31no—
32#1787—31no—
33#1787—31no—
34C22×SL(2,𝔽3)—21no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)Q812yes—
37C22×SL(2,𝔽3)C2311yescharacteristic
38C2×C3×C22⋊C4—41no—
39C2×#217—41no—
40C2×#217—41no—
41C2×C4⋊Dic3—41no—
42C2×Dic3⋊C4—41no—
43C2×Dic3⋊C4—41no—
44C4×SL(2,𝔽3)—21no—
45C4×SL(2,𝔽3)D412yes—
46#859—21no—
47#859—21no—
48#859—21no—
49#859—21no—
50C3⋊#95—41no—
51C23×Q8C2×D311yescharacteristic
52C2×#125—31no—
53#519—22no—
54#519—62no—
55C2×C4×Q8—31no—
56C2×C4×Q8—31no—
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×Q8C2×D311yescharacteristic
59C2×#117—31no—
60C2×#117—31no—
61#479—62no—
62#479—64no—
63#479—62no—
64C2×#107—31no—
65C2×#107—31no—
66C2×#106—31no—
67C2×#106—31no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#103—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—31no—
80#398—61no—
81#398—61no—
82#396—61no—
83#396—61no—
84#389—31no—
85#389—31no—
86#384—31no—
87#380—31no—
88#380—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#336—31no—
94#336—31no—
95C24×C3—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C2×SL(2,𝔽3)—22no—
103C2×SL(2,𝔽3)C2×D411yescharacteristic
104C2×SL(2,𝔽3)C2×Q811yescharacteristic
105C2×SL(2,𝔽3)#4111yescharacteristic
106C3×C22⋊C4—82no—
107#217—82no—
108#217—82no—
109C4⋊Dic3—82no—
110Dic3⋊C4—82no—
111Dic3⋊C4—82no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—31no—
115C22×Q8—32no—
116C22×Q8—61no—
117C22×Q8—61no—
118C22×Q8Dic612yes—
119C22×Q8C22×D311yescharacteristic
120C23×C4—31no—
121C22×C8—31no—
122C22×C8—31no—
123C22×C8—31no—
124C22×C8—31no—
125#125—62no—
126#125—62no—
127C4×Q8—21no—
128C4×Q8—32no—
129C4×Q8—34no—
130C4×Q8—62no—
131C4×Q8—62no—
132C4×Q8—62no—
133C4×Q8—61no—
134C4×Q8#6712yes—
135#117—62no—
136#117—62no—
137#117—62no—
138#117—62no—
139C2×C4⋊C4—31no—
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—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4S411yescharacteristic
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—61no—
160#107—62no—
161#107—62no—
162#106—62no—
163#106—62no—
164C4⋊C8—62no—
165C4⋊C8—62no—
166#103—61no—
167#103—61no—
168#103—62no—
169#103—62no—
170#103—61no—
171#103—62no—
172#103—61no—
173#103—62no—
174#103—61no—
175#103—61no—
176#103—61no—
177#103—61no—
178C22⋊C8—62no—
179C22⋊C8—62no—
180#95—61no—
181#95—61no—
182#95—61no—
183#95—61no—
184C23×C3—42no—
185C23×C3—41no—
186C23×C3—81no—
187C23×C3—81no—
188C23×C3—81no—
189C23×C3—81no—
190C23×C3—81no—
191C23×C3—81no—
192C2×C3×C4—42no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210SL(2,𝔽3)C22⋊Q811yescharacteristic
211C24C2×S411yescharacteristic
212C2×Q8—22no—
213C2×Q8—31no—
214C2×Q8—31no—
215C2×Q8—32no—
216C2×Q8—31no—
217C2×Q8—32no—
218C2×Q8—32no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—64no—
223C2×Q8—62no—
224C2×Q8—62no—
225C2×Q8—62no—
226C2×Q8C2×Dic611yescharacteristic
227C2×Q8#23511yescharacteristic
228C2×Q8C2×#6711yescharacteristic
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—32no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4C2×S411yescharacteristic
244C22×C4C2×S411yescharacteristic
245C2×C8—61no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C4⋊C4—32no—
258C4⋊C4—32no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—62no—
271C4⋊C4—122no—
272C4⋊C4—122no—
273C4⋊C4—241no—
274C4⋊C4—241no—
275C4⋊C4—241no—
276C4⋊C4—241no—
277C22⋊C4—22no—
278C22⋊C4—62no—
279C22⋊C4—241no—
280C22⋊C4—241no—
281C42—32no—
282C42—32no—
283C42—61no—
284C42—61no—
285C42—64no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3—82no—
296C22×C3—161no—
297C22×C3—161no—
298C22×C3—161no—
299C22×C3—161no—
300C3×C4—82no—
301C3×C4—161no—
302Dic3—82no—
303Dic3—82no—
304Dic3—161no—
305Dic3—161no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23A4⋊Q812yes—
313C23C22×S411yescharacteristic
314Q8—31no—
315Q8—31no—
316Q8—31no—
317Q8—34no—
318Q8—61no—
319Q8—62no—
320Q8—62no—
321Q8—62no—
322Q8#92411yescharacteristic
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4GL(2,ℤ/4)12yes—
360C8—121no—
361C8—121no—
362C8—121no—
363C8—121no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—162no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22C2×A4⋊Q811yescharacteristic
380C22#591211yescharacteristic
381C22#592611yescharacteristic
382C22#592611yescharacteristic
383C22#592711yescharacteristic
384C22#592711yescharacteristic
385C22C2×GL(2,ℤ/4)11yescharacteristic
386C4—22no—
387C4—31no—
388C4—31no—
389C4—31no—
390C4—31no—
391C4—41no—
392C4—62no—
393C4—62no—
394C4—121no—
395C4—241no—
396C4—241no—
397C4—241no—
398C4—241no—
399C3—41noSylow
400C2—42no—
401C2#8535111yescharacteristic
402C2#8535111yescharacteristic
403C2#8535211yescharacteristic
404C2#8535811yescharacteristic
405C2#8548011yescharacteristic
406C2#8553811yescharacteristic
407C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#519D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5389C2semidirect
5. #73062C2semidirect
6. C4⋊#859C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#519⋊C32
C2×#53892
#731482
#730622
C4⋊#8592
#730592
#329673
#49484