Finite Group Databasebeta

← #11713864

Subgroups — #11713864

Order 768.

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

normal subgroups only

This group has 471 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
1C2×#5419C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic, complemented
3C22⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
4#73148C211yesmaximal, characteristic
5#73063C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7#73061C211yesmaximal, characteristic, complemented
8#27242—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5419—22no—
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5389—22no—
18#4952—41nomaximal
19C2×#479—31no—
20C2×#451—31no—
21#2489—31no—
22Q8×C22⋊C4D311yescharacteristic
23#2077—31no—
24#2056—31no—
25#2036—31no—
26#2026—31no—
27#1971—31no—
28#1928—31no—
29#1924—31no—
30#1878—31no—
31#1878—31no—
32#1787—31no—
33#1763—31no—
34C22×SL(2,𝔽3)—21no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)—21no—
37C22×SL(2,𝔽3)C2311yescharacteristic
38C2×#226—21no—
39C2×#226—21no—
40C2×#226—21no—
41C2×C3×C22⋊C4—41no—
42C2×#217—41no—
43C2×#217—41no—
44C2×Dic3⋊C4—41no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)—21no—
47#859—21no—
48#859—21no—
49#859D412yes—
50C3⋊#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C23×Q8C2×D311yescharacteristic
54C22×Q16—31no—
55C22×Q16—31no—
56C2×#125—31no—
57C2×C22⋊Q8—31no—
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60#479—62no—
61#479—62no—
62#451—62no—
63#451—62no—
64C2×#107—31no—
65C2×#106—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×C22⋊C8—31no—
76C2×C22⋊C8—31no—
77C2×#97—31no—
78#400—61no—
79#397—61no—
80#391—31no—
81#391—31no—
82#391—31no—
83#391—31no—
84#388—61no—
85#382—61no—
86#380—31no—
87#380—31no—
88C4×C4⋊C4—31no—
89C4×C22⋊C4—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#340—31no—
94#336—31no—
95#336—31no—
96C24×C3—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)—41no—
104C2×SL(2,𝔽3)C2×D411yescharacteristic
105C2×SL(2,𝔽3)#4111yescharacteristic
106C2×SL(2,𝔽3)#4111yescharacteristic
107#226—41no—
108C3×C22⋊C4—82no—
109#217—82no—
110#217—82no—
111Dic3⋊C4—82no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—31no—
116C22×Q8—31no—
117C22×Q8—61no—
118C22×Q8—61no—
119C22×Q8—61no—
120C22×Q8C22×D311yescharacteristic
121C23×C4—31no—
122C2×Q16—62no—
123C2×Q16—62no—
124C2×Q16—61no—
125C2×Q16—61no—
126C2×Q16—61no—
127C2×Q16—61no—
128C2×Q16—61no—
129C2×Q16—61no—
130C2×Q16—121no—
131C22×C8—31no—
132C22×C8—31no—
133C22×C8—31no—
134C22×C8—31no—
135#125—62no—
136#125—121no—
137C22⋊Q8—62no—
138C22⋊Q8—121no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—61no—
142C4×Q8—61no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4S411yescharacteristic
155C2×C42—31no—
156C2×C42—31no—
157C2×C42—61no—
158#107—62no—
159#106—62no—
160C4⋊C8—62no—
161#103—32no—
162#103—32no—
163#103—62no—
164#103—61no—
165#103—62no—
166#103—61no—
167#103—61no—
168#103—61no—
169#103—121no—
170#103—121no—
171#103—121no—
172C22⋊C8—62no—
173C22⋊C8—121no—
174#97—121no—
175#95—31no—
176#95—31no—
177#95—61no—
178#95—61no—
179#95—61no—
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—81no—
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—81no—
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)#12411yescharacteristic
211C24C2×S411yescharacteristic
212C2×Q8—31no—
213C2×Q8—31no—
214C2×Q8—31no—
215C2×Q8—32no—
216C2×Q8—41no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×Q8—122no—
229C2×Q8—121no—
230C2×Q8—121no—
231C2×Q8—121no—
232C2×Q8#23511yescharacteristic
233C2×Q8D3×D411yescharacteristic
234C2×Q8#23711yescharacteristic
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4C2×S411yescharacteristic
250C22×C4C2×S411yescharacteristic
251Q16—124no—
252Q16—121no—
253Q16—121no—
254Q16—121no—
255Q16—121no—
256C2×C8—32no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C4⋊C4—32no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—122no—
279C4⋊C4—122no—
280C4⋊C4—122no—
281C22⋊C4—22no—
282C22⋊C4—121no—
283C22⋊C4—122no—
284C22⋊C4—122no—
285C42—61no—
286C42—61no—
287C42—61no—
288C42—61no—
289C42—122no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—82no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—161no—
301C22×C3—161no—
302C22×C3—161no—
303C22×C3—161no—
304C3×C4—161no—
305C3×C4—161no—
306Dic3—82no—
307Dic3—161no—
308Dic3—161no—
309Dic3—161no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23#98312yes—
317C23C22×S411yescharacteristic
318Q8—31no—
319Q8—31no—
320Q8—31no—
321Q8—61no—
322Q8—62no—
323Q8—62no—
324Q8—121no—
325Q8—121no—
326Q8—121no—
327Q8—121no—
328Q8—121no—
329Q8—121no—
330Q8#88511yescharacteristic
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—62no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C8—62no—
369C8—121no—
370C8—121no—
371C8—121no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—162no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22#591211yescharacteristic
388C22D4×S411yescharacteristic
389C22#591411yescharacteristic
390C22C2×#98311yescharacteristic
391C22#592311yescharacteristic
392C22#592711yescharacteristic
393C22#592711yescharacteristic
394C4—31no—
395C4—31no—
396C4—31no—
397C4—31no—
398C4—41no—
399C4—41no—
400C4—121no—
401C4—121no—
402C4—121no—
403C4—122no—
404C4—241no—
405C4—241no—
406C4—241no—
407C3—41noSylow
408C2—42no—
409C2#8530011yescharacteristic
410C2#8533011yescharacteristic
411C2#8535211yescharacteristic
412C2#8537811yescharacteristic
413C2#8538311yescharacteristic
414C2#8539111yescharacteristic
415C2#8539411yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C22⋊C4D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#124
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. #73063C2semidirect
6. #73061C2semidirect
7. #73061C2semidirect

Maximal subgroups

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

subgroupindex
C2×#54192
C2×#53892
C22⋊C4×SL(2,𝔽3)2
#731482
#730632
#730612
#272423
#49524