Finite Group Databasebeta

← #11712909

Subgroups — #11712909

Order 768.

624 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 81 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⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#5426C211yesmaximal, characteristic, complemented
3C2×#5424C211yesmaximal, characteristic, complemented
4#73151C211yesmaximal, characteristic, complemented
5#73151C211yesmaximal, characteristic, complemented
6#73064C211yesmaximal, characteristic, complemented
7#73063C211yesmaximal, characteristic, complemented
8#22422—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×#41⋊C3C2211yescharacteristic
11C4⋊(#41⋊C3)C414yescomplemented
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5426—22no—
14#5424—22no—
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18C2×#859C2211yescharacteristic
19#4980—41nomaximal
20C2×#550D311yescharacteristic
21C2×#435—31no—
22C2×#417—31no—
23C2×#416—31no—
24#2352—31no—
25#1982—31no—
26#1961—31no—
27#1895—31no—
28#1893—31no—
29#1875—31no—
30#1874—31no—
31#1805—31no—
32#1788—31no—
33#1788—31no—
34#1763—31no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3—21no—
39C2×#41⋊C3C2×C412yes—
40C2×#41⋊C3C2×C412yescomplemented
41C22×SL(2,𝔽3)C2311yescharacteristic
42C2×C3×C4⋊C4—41no—
43C2×C4⋊Dic3—41no—
44C2×C4×Dic3—41no—
45C4×SL(2,𝔽3)C2×C412yescomplemented
46#859—21no—
47#859—21no—
48#859—21no—
49#859D412yescomplemented
50C3⋊#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C3⋊#95—41no—
54C22×#41C2×D311yescharacteristic
55C22×#41C2×D311yescharacteristic
56C22×M16—31no—
57C23×C8—31no—
58#550—34no—
59#550Dic314yescomplemented
60C2×#127—31no—
61C2×#121—31no—
62C2×#121—31no—
63C2×C4×Q8C2×D311yescharacteristic
64C2×C4×D4—31no—
65C2×#117—31no—
66C22×C4⋊C4—31no—
67#435—64no—
68C2×C4⋊C8—31no—
69#417—62no—
70#417—62no—
71#416—62no—
72#416—62no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×#102—31no—
80C2×#102—31no—
81C2×#102—31no—
82C2×#102—31no—
83C2×#102—31no—
84C2×#102—31no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8—31no—
87C2×#97—31no—
88#388—61no—
89#387—61no—
90#386—61no—
91#382—31no—
92#380—31no—
93#380—31no—
94C4×C4⋊C4—31no—
95C4×C22⋊C4—61no—
96#340—31no—
97#340—31no—
98#340—31no—
99#340—31no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107#41⋊C3—22no—
108#41⋊C3—22no—
109C2×SL(2,𝔽3)C22×C411yescharacteristic
110C2×SL(2,𝔽3)C2×D411yescharacteristic
111C2×SL(2,𝔽3)#4111yescharacteristic
112C3×C4⋊C4—44no—
113C4⋊Dic3—82no—
114C4×Dic3—82no—
115C2×#41—21no—
116C2×#41—21no—
117C2×#41—21no—
118C2×#41—21no—
119C2×#41—32no—
120C2×#41—32no—
121C2×#41—61no—
122C2×#41—61no—
123C2×#41—61no—
124C2×#41—61no—
125C2×#41C2×Dic312yes—
126C2×#41C2×Dic312yes—
127C22×Q8C22×D311yescharacteristic
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131C22×D4—61no—
132C23×C4—31no—
133C23×C4—31no—
134C23×C4—61no—
135C2×M16—62no—
136C2×M16—62no—
137C2×M16—121no—
138C22×C8—31no—
139C22×C8—31no—
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—62no—
143C22×C8—62no—
144C22×C8—121no—
145#127—34no—
146#127—62no—
147#121—34no—
148#121—34no—
149#121—62no—
150#121—62no—
151C4×Q8—32no—
152C4×Q8C2×Dic312yes—
153C4×D4—34no—
154C4×D4—34no—
155#117—62no—
156#117—121no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—121no—
165C2×C4⋊C4S411yescharacteristic
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C42—31no—
172C2×C42—61no—
173C2×C42—61no—
174C4⋊C8—62no—
175#103—32no—
176#103—32no—
177#103—61no—
178#103—61no—
179#103—61no—
180#103—61no—
181#103—61no—
182#103—61no—
183#102—62no—
184#102—62no—
185#102—62no—
186#102—62no—
187#102—121no—
188#102—121no—
189#102—121no—
190#102—121no—
191C22⋊C8—62no—
192C22⋊C8—62no—
193#97—62no—
194#95—61no—
195#95—61no—
196#95—61no—
197#95—61no—
198#95—61no—
199C23×C3—41no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221SL(2,𝔽3)C4×D411yescharacteristic
222C24—31no—
223C24—31no—
224#41—22no—
225#41—22no—
226#41—62no—
227#41—62no—
228#41—62no—
229#41—62no—
230#41—62no—
231#41—62no—
232#41—122no—
233#41—122no—
234C2×Q8—31no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8D3×D411yescharacteristic
238C2×Q8#23711yescharacteristic
239C2×Q8C22×Dic311yescharacteristic
240C2×D4—32no—
241C2×D4—32no—
242C2×D4—34no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—64no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—241no—
257C2×D4—241no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—122no—
277C22×C4—122no—
278C22×C4—241no—
279C22×C4C2×S411yescharacteristic
280C22×C4C2×S411yescharacteristic
281C22×C4C2×S411yescharacteristic
282M16—124no—
283C2×C8—32no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—124no—
296C4⋊C4—32no—
297C4⋊C4—32no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—122no—
307C4⋊C4SL(2,ℤ/4)14yes—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—122no—
311C22⋊C4—122no—
312C22⋊C4—241no—
313C42—32no—
314C42—32no—
315C42—122no—
316C42—241no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C3×C4—82no—
325C3×C4—82no—
326C3×C4—82no—
327Dic3—82no—
328Dic3—82no—
329Dic3—161no—
330Dic3—161no—
331C23—32no—
332C23—32no—
333C23—61no—
334C23—61no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23C22×S411yescharacteristic
346Q8—31no—
347Q8—31no—
348Q8—31no—
349Q8—61no—
350Q8D4×Dic311yescharacteristic
351D4—62no—
352D4—62no—
353D4—62no—
354D4—62no—
355D4—124no—
356D4—121no—
357D4—121no—
358D4—121no—
359D4—121no—
360D4—244no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—32no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—62no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—244no—
397C2×C4C2×SL(2,ℤ/4)12yes—
398C2×C4C2×SL(2,ℤ/4)12yes—
399C2×C4C2×SL(2,ℤ/4)12yes—
400C8—62no—
401C8—121no—
402C8—121no—
403C8—121no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—41no—
411C22—62no—
412C22—62no—
413C22—62no—
414C22—62no—
415C22—62no—
416C22—62no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22D4×S411yescharacteristic
426C22#591411yescharacteristic
427C22#592311yescharacteristic
428C22#592311yescharacteristic
429C22#592411yescharacteristic
430C22#592511yescharacteristic
431C22C22×SL(2,ℤ/4)11yescharacteristic
432C4—22no—
433C4—22no—
434C4—22no—
435C4—31no—
436C4—31no—
437C4—31no—
438C4—31no—
439C4—62no—
440C4—122no—
441C4—241no—
442C4—241no—
443C4—241no—
444C3—41noSylow
445C2—122no—
446C2—122no—
447C2#8540511yescharacteristic
448C2#8540911yescharacteristic
449C2#8541111yescharacteristic
450C2#8541311yescharacteristic
451C2D4×SL(2,ℤ/4)11yescharacteristic
452C2#8551911yescharacteristic
453C2#8552811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
cosocleC22×SL(2,𝔽3)C23
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C4×D4
LCSG ▷ C2×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
2. C2×#5426C2semidirect
3. C2×#5424C2semidirect
4. #73151C2semidirect
5. #73151C2semidirect
6. #73064C2semidirect
7. #73063C2semidirect
11. C4⋊(#41⋊C3)C4semidirect
40. C2×#41⋊C3C2×C4semidirect
45. C4×SL(2,𝔽3)C2×C4semidirect
49. #859D4semidirect
59. #550Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#54262
C2×#54242
#731512
#730642
#730632
#224223
#49804