Finite Group Databasebeta

← #11711981

Subgroups — #11711981

Order 768.

504 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#41×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#85325C211yesmaximal, characteristic, complemented
3#73100C211yesmaximal, characteristic, complemented
4#73099C211yesmaximal, characteristic, complemented
5#73088C211yesmaximal, characteristic, complemented
6#73087C211yesmaximal, characteristic, complemented
7C2×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
8#18518—31nomaximal, Sylow
9C2×#950—41nomaximal
10Q8×SL(2,𝔽3)C2211yescharacteristic
11D4×SL(2,𝔽3)—21no—
12D4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)—21no—
14C2×C4×SL(2,𝔽3)C2211yescharacteristic
15#5421—21no—
16C4⋊GL(2,𝔽3)C2211yescharacteristic
17C4×GL(2,𝔽3)—21no—
18#5386C2211yescharacteristic
19SL(2,𝔽3)⋊C8C2211yescharacteristic
20SL(2,𝔽3)⋊C8C2211yescharacteristic
21Q8×#41D311yescharacteristic, complemented
22#3167—31no—
23#3103—31no—
24#3035—31no—
25#1653—31no—
26#1651—31no—
27#1645—31no—
28#1643—31no—
29#1627—31no—
30#1626—31no—
31#1623—31no—
32#1622—31no—
33#1542—31no—
34#1476—31no—
35C2×Q8⋊C8—31no—
36C2×C3×#41—41no—
37C2×#235—41no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)D411yescharacteristic
40C2×GL(2,𝔽3)—21no—
41#950—48no—
42C2×#216—41no—
43C2×#215—41no—
44C2×#214—41no—
45C2×#213—41no—
46C22×C3⋊C8—41no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)D411yescharacteristic
49C4×SL(2,𝔽3)C2311yescharacteristic
50#859—21no—
51C2×#135—61no—
52Q82C2×D311yescharacteristic
53#557—31no—
54#554—31no—
55#554—61no—
56D4×Q8—21no—
57D4×Q8C2×D311yescharacteristic
58#533—31no—
59#533—61no—
60#532—31no—
61C4×#41—31no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×D311yescharacteristic
64#498—31no—
65#492—31no—
66#487—31no—
67#486—31no—
68#481—61no—
69#475—31no—
70#475—31no—
71#465—62no—
72#464—61no—
73#463—61no—
74#462—31no—
75#461—31no—
76#460—31no—
77#460—31no—
78#460—31no—
79#459—31no—
80#448—61no—
81C4×SD16—61no—
82C4×SD16—62no—
83C4×SD16—61no—
84#431—31no—
85C2×C4⋊C8—31no—
86#416—61no—
87C2×C4×C8—31no—
88#333—31no—
89#333—31no—
90#332—31no—
91#332—31no—
92Q8⋊C8—31no—
93Q8⋊C8—31no—
94Q8⋊C8—31no—
95Q8⋊C8—31no—
96Q8⋊C8—31no—
97Q8⋊C8—31no—
98D4⋊C8—31no—
99D4⋊C8—31no—
100C3×#41—44no—
101C3×#41—82no—
102C2×C3×Q8—41no—
103C2×C3×D4—41no—
104C2×C3×D4—81no—
105C22×C3×C4—41no—
106C22×C3×C4—81no—
107C2×#67—81no—
108#235—44no—
109#235—82no—
110C2×D12—41no—
111C2×C4×D3—81no—
112C2×Dic6—41no—
113C2×SL(2,𝔽3)—21no—
114C2×SL(2,𝔽3)—41no—
115C2×SL(2,𝔽3)C2×D411yescharacteristic
116GL(2,𝔽3)—41no—
117#216—44no—
118#215—44no—
119#214—44no—
120#213—44no—
121C2×C3⋊C8—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8—44no—
124C2×#41—61no—
125C2×#41S411yescharacteristic
126C22×Q8—21no—
127C22×Q8#6711yescharacteristic
128#135—124no—
129C2×Q16—61no—
130C2×SD16—61no—
131C2×SD16—62no—
132C2×SD16—61no—
133C2×SD16—61no—
134C2×SD16—61no—
135C2×D8—61no—
136C22×C8—61no—
137C22×C8—61no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8—31no—
141C4⋊Q8—61no—
142C4⋊Q8—61no—
143#127—31no—
144#125—61no—
145#125—62no—
146#124—61no—
147C22⋊Q8—32no—
148C22⋊Q8—62no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152#121—61no—
153#121—61no—
154C4×Q8—21no—
155C4×Q8—31no—
156C4×Q8—31no—
157C4×Q8—32no—
158C4×Q8—31no—
159C4×Q8—31no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×Q8#6711yescharacteristic
163C4×Q8C22×D311yescharacteristic
164C4×D4—31no—
165C4×D4—31no—
166C4×D4—61no—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170#117—31no—
171#117—61no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C42—31no—
175C2×C42—61no—
176#106—62no—
177#106—61no—
178#106—61no—
179C4⋊C8—31no—
180C4⋊C8—31no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—61no—
184#103—61no—
185#103—61no—
186#103—61no—
187#103—62no—
188#103—61no—
189#102—61no—
190#102—61no—
191#102—61no—
192#102—61no—
193#102—62no—
194C22⋊C8—61no—
195C4×C8—31no—
196C4×C8—31no—
197C4×C8—61no—
198C4×C8—61no—
199C23×C3—41no—
200C23×C3—81no—
201C22×D3—81no—
202C3×Q8—42no—
203C3×Q8—81no—
204C3×D4—42no—
205C3×D4—81no—
206C3×D4—84no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—84no—
212C2×C3×C4—81no—
213#67—84no—
214C2×Dic3—81no—
215D12—42no—
216D12—81no—
217C4×D3—84no—
218Dic6—42no—
219Dic6—81no—
220SL(2,𝔽3)#13511yescharacteristic, complemented
221C3⋊C8—42no—
222C3⋊C8—42no—
223#41—22no—
224#41—122no—
225#41—122no—
226#41GL(2,𝔽3)14yescomplemented
227C2×Q8—21no—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8—41no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—121no—
235C2×Q8C2×#6711yescharacteristic
236C2×Q8C2×S411yescharacteristic
237C2×D4—21no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C2×D4C2×S411yescharacteristic
243C22×C4—21no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—121no—
249C22×C4C2×S411yescharacteristic
250Q16—122no—
251SD16—124no—
252SD16—121no—
253SD16—122no—
254SD16—121no—
255SD16—122no—
256SD16—122no—
257D8—122no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—122no—
265C2×C8—122no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C42—31no—
285C42—31no—
286C42—31no—
287C42—31no—
288C42—61no—
289C42—61no—
290C42—61no—
291C22×C3—41no—
292C22×C3—42no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—82no—
296C22×C3—161no—
297C22×C3—161no—
298C2×D3—82no—
299C2×D3—161no—
300C2×D3—161no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—42no—
304C3×C4—82no—
305Dic3—82no—
306C23—21no—
307C23—121no—
308C23GL(2,ℤ/4)11yescharacteristic
309Q8—21no—
310Q8—31no—
311Q8—61no—
312Q8—61no—
313Q8—121no—
314Q8—121no—
315Q8—122no—
316Q8—121no—
317Q8—121no—
318Q8#95011yescharacteristic, complemented
319Q8C2×GL(2,𝔽3)12yes—
320D4—24no—
321D4—21no—
322D4—121no—
323D4—121no—
324D4—122no—
325D4—121no—
326D4—121no—
327D4—242no—
328D4C2×GL(2,𝔽3)12yes—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—241no—
344C2×C4—242no—
345C2×C4C2×GL(2,𝔽3)12yes—
346C2×C4#98312yes—
347C2×C4GL(2,ℤ/4)11yescharacteristic
348C2×C4C22×S411yescharacteristic
349C8—62no—
350C8—121no—
351C8—121no—
352C8—121no—
353C8—121no—
354C8—121no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—82no—
359C2×C3—162no—
360D3—162no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—241no—
367C22—241no—
368C22—241no—
369C22#542112yes—
370C22C2×GL(2,ℤ/4)11yescharacteristic
371C4—22no—
372C4—31no—
373C4—31no—
374C4—61no—
375C4—61no—
376C4—61no—
377C4—121no—
378C4—121no—
379C4—241no—
380C4#542112yes—
381C4C22×GL(2,𝔽3)11yescharacteristic
382C4C2×#98311yescharacteristic
383C3—41noSylow
384C2—22no—
385C2—42no—
386C2—481no—
387C2C2×#542111yescharacteristic
388C2#8551411yescharacteristic
389C2#8554411yescharacteristic
subgroupG/H
centerC2×C4GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C23
FrattiniC2×C4C22×S4
FittingQ8×#41D3
socleC22C2×GL(2,ℤ/4)
hypercenterC2×#41S4
hypocenterSL(2,𝔽3)#135
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #41×SL(2,𝔽3)C2semidirect
2. #85325C2semidirect
3. #73100C2semidirect
4. #73099C2semidirect
5. #73088C2semidirect
6. #73087C2semidirect
7. C2×SL(2,𝔽3)⋊C8C2semidirect
21. Q8×#41D3semidirect
220. SL(2,𝔽3)#135semidirect
226. #41GL(2,𝔽3)semidirect
318. Q8#950semidirect

Maximal subgroups

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

subgroupindex
#41×SL(2,𝔽3)2
#853252
#731002
#730992
#730882
#730872
C2×SL(2,𝔽3)⋊C82
#185183
C2×#9504