Finite Group Databasebeta

← #11711801

Subgroups — #11711801

Order 768.

441 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×2−1+4⋊C3C211yesmaximal, characteristic
2#73072C211yesmaximal, characteristic, complemented
3#73069C211yesmaximal, characteristic, complemented
4#73057C211yesmaximal, characteristic, complemented
5#73056C211yesmaximal, characteristic, complemented
6#73055C211yesmaximal, characteristic, complemented
7C2×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
8#17454—31nomaximal, Sylow
9C2×2−1+4⋊C3C411yescharacteristic
10#520⋊C3C411yescharacteristic
11#519⋊C3C411yescharacteristic
12#519⋊C3C411yescharacteristic
13C4×#41⋊C3C2211yescharacteristic
14C2×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×C4×SL(2,𝔽3)C2211yescharacteristic
16#5011—41nomaximal
17SL(2,𝔽3)⋊C8C2211yescharacteristic
18SL(2,𝔽3)⋊C8C2211yescharacteristic
19SL(2,𝔽3)⋊C8C2211yescharacteristic
20SL(2,𝔽3)⋊C8C2211yescharacteristic
21C4×2−1+4D311yescharacteristic
22#2989—31no—
23#2965—31no—
24#1491—31no—
25#1491—31no—
26#1489—31no—
27#1489—31no—
28#1488—31no—
29#1488—31no—
30#1487—31no—
31#1487—31no—
32#1479—31no—
33#1477—31no—
34#1476—31no—
35C2×Q8⋊C8—31no—
362−1+4⋊C3—22no—
37C2×#41⋊C3C2×C411yescharacteristic
38C22×SL(2,𝔽3)C2×C411yescharacteristic
39C22×SL(2,𝔽3)C2×C411yescharacteristic
40C3×C4×D4—41no—
41C2×#208—41no—
42C22×C3⋊C8—41no—
43C4×SL(2,𝔽3)C2×C411yescharacteristic
44C4×SL(2,𝔽3)C2×C411yescharacteristic
45C4×SL(2,𝔽3)C2×C411yescharacteristic
46C4×SL(2,𝔽3)C2311yescharacteristic
47#830—41no—
48#830—41no—
49C4⋊(C3⋊C8)—41no—
50#803—41no—
51C2×2−1+4Dic311yescharacteristic
52#520—31no—
53#520—31no—
54#520—31no—
55#520Dic311yescharacteristic
56#519—31no—
57#519Dic311yescharacteristic
58#519Dic311yescharacteristic
59C4×#41—31no—
60C4×#41—31no—
61C4×#41—31no—
62C4×#41C2×D311yescharacteristic
63C2×C4×Q8—31no—
64C2×C4×Q8C2×D311yescharacteristic
65C2×C4×Q8C2×D311yescharacteristic
66#446—61no—
67#446—61no—
68#436—61no—
69#436—61no—
70#435—61no—
71#435—61no—
72#435—61no—
73#435—61no—
74#432—31no—
75#432—31no—
76#431—31no—
77#431—31no—
78#431—31no—
79#431—31no—
80#423—31no—
81#423—31no—
82C2×C4⋊C8—31no—
83C4×M16—31no—
84C4×M16—31no—
85C2×C4×C8—31no—
86Q8⋊C8—31no—
87Q8⋊C8—31no—
88Q8⋊C8—31no—
89Q8⋊C8—31no—
90Q8⋊C8—31no—
91Q8⋊C8—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—
100D4⋊C8—31no—
101D4⋊C8—31no—
102C2×C3×D4—41no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105#41⋊C3—21no—
106C2×SL(2,𝔽3)—21no—
107C2×SL(2,𝔽3)M1612yescomplemented
108C2×SL(2,𝔽3)C22×C411yescharacteristic
109C3×C4⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C42—41no—
113#208—82no—
114C2×C3⋊C8—41no—
115C2×C3⋊C8—41no—
116C2×C3⋊C8—41no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8—82no—
1192−1+4—22no—
1202−1+4—62no—
121C2×#41—31no—
122C2×#41—31no—
123C2×#41—31no—
124C2×#41C2×Dic311yescharacteristic
125C22×Q8—31no—
126C22×Q8C2×Dic311yescharacteristic
127C22×Q8C2×Dic311yescharacteristic
128C2×M16—61no—
129C2×M16—61no—
130C2×M16—61no—
131C2×M16—61no—
132C22×C8—61no—
133C22×C8—61no—
134C4×Q8—31no—
135C4×Q8—31no—
136C4×Q8—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×Q8—31no—
142C4×Q8—31no—
143C4×Q8—31no—
144C4×Q8—61no—
145C4×Q8C2×Dic311yescharacteristic
146C4×Q8C2×Dic311yescharacteristic
147C4×Q8C2×Dic311yescharacteristic
148C4×Q8C22×D311yescharacteristic
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—31no—
152C4×D4—31no—
153C4×D4—31no—
154C4×D4—31no—
155C4×D4—31no—
156C4×D4—31no—
157C4×D4—31no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4S411yescharacteristic
161#117—31no—
162#117—31no—
163#117—31no—
164#117—31no—
165#117—31no—
166#117—31no—
167#117—61no—
168#117—61no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C42—31no—
174C2×C42—31no—
175C2×C42—31no—
176C2×C42—61no—
177C4⋊C8—31no—
178C4⋊C8—31no—
179C4⋊C8—31no—
180C4⋊C8—31no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#97—61no—
194#97—61no—
195#97—61no—
196#97—61no—
197C4×C8—31no—
198C4×C8—31no—
199C4×C8—31no—
200C4×C8—31no—
201C4×C8—61no—
202C4×C8—61no—
203C23×C3—41no—
204C23×C3—41no—
205C3×D4—82no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215SL(2,𝔽3)C2×M1611yescharacteristic
216C3⋊C8—42no—
217C3⋊C8—81no—
218C3⋊C8—81no—
219C3⋊C8—81no—
220#41—21no—
221#41—34no—
222#41—62no—
223#41—62no—
224#41—62no—
225#41—62no—
226#41—62no—
227#41—61no—
228C2×Q8—21no—
229C2×Q8—31no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—32no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8—61no—
236C2×Q8#20812yescomplemented
237C2×Q8C22×Dic311yescharacteristic
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—31no—
241C2×D4SL(2,ℤ/4)11yescharacteristic
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4C2×S411yescharacteristic
252C22×C4C2×S411yescharacteristic
253M16—122no—
254M16—122no—
255M16—241no—
256M16—241no—
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—122no—
266C2×C8—241no—
267C4⋊C4—31no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—31no—
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4SL(2,ℤ/4)11yescharacteristic
279C22⋊C4—31no—
280C22⋊C4—31no—
281C22⋊C4—31no—
282C22⋊C4—31no—
283C22⋊C4—61no—
284C22⋊C4SL(2,ℤ/4)11yescharacteristic
285C22⋊C4SL(2,ℤ/4)11yescharacteristic
286C42—31no—
287C42—31no—
288C42—31no—
289C42—31no—
290C42—31no—
291C42—31no—
292C42—31no—
293C42—31no—
294C42—31no—
295C42—61no—
296C42—61no—
297C42C2×S411yescharacteristic
298C22×C3—42no—
299C22×C3—41no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—81no—
311C23—31no—
312C23C2×SL(2,ℤ/4)11yescharacteristic
313C23C2×SL(2,ℤ/4)11yescharacteristic
314Q8—32no—
315Q8—31no—
316Q8—32no—
317Q8—62no—
318Q8—61no—
319Q8—61no—
320Q8C2×#20811yescharacteristic
321D4—22no—
322D4—32no—
323D4—62no—
324D4—61no—
325D4—62no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—64no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—241no—
350C2×C4C2×SL(2,ℤ/4)11yescharacteristic
351C2×C4C2×SL(2,ℤ/4)11yescharacteristic
352C2×C4C2×SL(2,ℤ/4)11yescharacteristic
353C2×C4C2×SL(2,ℤ/4)11yescharacteristic
354C2×C4C22×S411yescharacteristic
355C8—62no—
356C8—121no—
357C8—121no—
358C8—121no—
359C8—121no—
360C8—121no—
361C8—121no—
362C8—121no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—82no—
367C2×C3—161no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22#540912yes—
377C22C22×SL(2,ℤ/4)11yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—31no—
383C4—31no—
384C4—32no—
385C4—61no—
386C4—61no—
387C4—61no—
388C4—61no—
389C4—61no—
390C4#592611yescharacteristic
391C4#592711yescharacteristic
392C3—41noSylow
393C2—22no—
394C2—41no—
395C2—121no—
396C2C2×#540911yescharacteristic
397C2#8553411yescharacteristic
398C2#8554111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC4×2−1+4D3
socleC22C22×SL(2,ℤ/4)
prime radicalC2×2−1+4⋊C3C4
cosocleC4×SL(2,𝔽3)C23
hypercenterC4×D4S4
hypocenterSL(2,𝔽3)C2×M16
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #73072C2semidirect
3. #73069C2semidirect
4. #73057C2semidirect
5. #73056C2semidirect
6. #73055C2semidirect
7. C2×SL(2,𝔽3)⋊C8C2semidirect
107. C2×SL(2,𝔽3)M16semidirect
236. C2×Q8#208semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4⋊C32
#730722
#730692
#730572
#730562
#730552
C2×SL(2,𝔽3)⋊C82
#174543
#50114