Finite Group Databasebeta

← #11713122

Subgroups — #11713122

Order 768.

638 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 67 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×#5394C211yesmaximal, characteristic, complemented
2C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C2×C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#2351⋊C3C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73065C211yesmaximal, characteristic, complemented
7#73063C211yesmaximal, characteristic, complemented
8#23010—31nomaximal, Sylow
9C22×GL(2,𝔽3)C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#5394—22no—
17C4⋊GL(2,𝔽3)C414yescomplemented
18C4×GL(2,𝔽3)—22no—
19#4665—41nomaximal
20C2×#460—31no—
21C2×#440—31no—
22C2×C4×SD16—31no—
23#2366—31no—
24#2351D311yescharacteristic, complemented
25#1961—31no—
26#1949—31no—
27#1936—31no—
28#1934—31no—
29#1880—31no—
30#1880—31no—
31#1864—31no—
32#1862—31no—
33#1771—31no—
34#1763—31no—
35C22×D12—41no—
36C22×C4×D3—41no—
37C22×SL(2,𝔽3)C2311yescharacteristic
38C2×GL(2,𝔽3)—21no—
39C2×GL(2,𝔽3)—21no—
40C2×GL(2,𝔽3)—21no—
41C2×GL(2,𝔽3)—21no—
42C2×GL(2,𝔽3)C2×C412yescomplemented
43C2×GL(2,𝔽3)C2×C412yescomplemented
44C2×(C2×D3)⋊C4—41no—
45C2×(C2×D3)⋊C4—41no—
46C2×(C2×D3)⋊C4—41no—
47C4×SL(2,𝔽3)—21no—
48C4×SL(2,𝔽3)—21no—
49C4×SL(2,𝔽3)C2×C412yescomplemented
50#859—21no—
51#859D412yescomplemented
52C3×#95—41no—
53C3⋊#95—41no—
54C22×SD16—31no—
55C22×SD16—31no—
56C2×#127—31no—
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×D4—31no—
61C2×C4×D4—31no—
62C2×C4×D4—61no—
63#460—34no—
64#460—34no—
65#440—62no—
66#440—62no—
67C4×SD16—62no—
68C4×SD16—62no—
69C2×#107—31no—
70C2×#106—31no—
71C2×C4⋊C8—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#102—31no—
75C2×#102—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×#102—31no—
80C2×#97—31no—
81C2×#97—31no—
82C2×C4×C8—31no—
83#390—31no—
84#390—31no—
85#390—61no—
86#389—31no—
87#389—31no—
88#386—61no—
89#386—61no—
90#382—31no—
91C4×C4⋊C4—31no—
92C4×C4⋊C4—31no—
93#340—31no—
94#340—31no—
95#340—31no—
96#340—31no—
97#336—31no—
98#336—31no—
99C23×D3—41no—
100C23×D3—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C2×D12—44no—
107C2×D12—82no—
108C2×D12—82no—
109C2×C4×D3—82no—
110C2×C4×D3—82no—
111C2×C4×D3—82no—
112C2×SL(2,𝔽3)C22×C411yescharacteristic
113C2×SL(2,𝔽3)C2×D411yescharacteristic
114C2×SL(2,𝔽3)#4111yescharacteristic
115GL(2,𝔽3)—22no—
116GL(2,𝔽3)—22no—
117(C2×D3)⋊C4—82no—
118(C2×D3)⋊C4—82no—
119(C2×D3)⋊C4—82no—
120C22×Q8C22×D311yescharacteristic
121C22×D4—31no—
122C22×D4—31no—
123C22×D4—61no—
124C22×D4—61no—
125C23×C4—61no—
126C23×C4—61no—
127C2×SD16—32no—
128C2×SD16—32no—
129C2×SD16—32no—
130C2×SD16—32no—
131C2×SD16—61no—
132C2×SD16—61no—
133C2×SD16—61no—
134C2×SD16—61no—
135C2×SD16—61no—
136C2×SD16—61no—
137C2×SD16—61no—
138C2×SD16—61no—
139C22×C8—31no—
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—31no—
143#127—34no—
144#127—62no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—32no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8C4×D312yescomplemented
151C4×D4—62no—
152C4×D4—62no—
153C4×D4—121no—
154C4×D4—121no—
155C4×D4—124no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42—31no—
172C2×C42—61no—
173#107—62no—
174#106—62no—
175C4⋊C8—34no—
176#103—32no—
177#103—32no—
178#103—61no—
179#103—61no—
180#102—34no—
181#102—34no—
182#102—62no—
183#102—62no—
184#102—121no—
185#102—121no—
186#97—62no—
187#97—62no—
188C4×C8—62no—
189#95—31no—
190#95—61no—
191#95—61no—
192#95S411yescharacteristic
193C23×C3—41no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—42no—
197C22×D3—42no—
198C22×D3—81no—
199C22×D3—81no—
200C22×D3—81no—
201C22×D3—81no—
202C22×D3—81no—
203C22×D3—81no—
204C22×D3—81no—
205C22×D3—81no—
206C22×D3—81no—
207C22×D3—81no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×Dic3—42no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223D12—88no—
224C4×D3—88no—
225SL(2,𝔽3)C4×D411yescharacteristic, complemented
226C24—61no—
227C24—61no—
228C2×Q8—31no—
229C2×Q8—31no—
230C2×Q8—31no—
231C2×Q8C2×C4×D311yescharacteristic
232C2×Q8D3×D411yescharacteristic
233C2×Q8#23911yescharacteristic
234C2×D4—32no—
235C2×D4—32no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—64no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—64no—
248C2×D4—122no—
249C2×D4—122no—
250C2×D4—241no—
251C2×D4—241no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—122no—
259C22×C4—122no—
260C22×C4—122no—
261C22×C4—122no—
262C22×C4—241no—
263C22×C4—241no—
264C22×C4C2×S411yescharacteristic
265C22×C4C2×S411yescharacteristic
266C22×C4C2×S411yescharacteristic
267SD16—62no—
268SD16—62no—
269SD16—62no—
270SD16—62no—
271SD16—62no—
272SD16—62no—
273SD16—62no—
274SD16—62no—
275C2×C8—32no—
276C2×C8—32no—
277C2×C8—32no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C4⋊C4—32no—
288C4⋊C4—32no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—62no—
298C4⋊C4—121no—
299C22⋊C4—122no—
300C22⋊C4—122no—
301C22⋊C4—122no—
302C22⋊C4—122no—
303C22⋊C4—241no—
304C22⋊C4—241no—
305C42—32no—
306C42—32no—
307C42—61no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42—122no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C2×D3—44no—
320C2×D3—82no—
321C2×D3—82no—
322C2×D3—82no—
323C2×D3—82no—
324C2×D3—82no—
325C2×D3—82no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—161no—
330C2×D3—161no—
331C2×D3—161no—
332C2×D3—161no—
333C3×C4—82no—
334C3×C4—82no—
335C3×C4—161no—
336Dic3—82no—
337Dic3—161no—
338C23—62no—
339C23—62no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23C22×S411yescharacteristic
353Q8—31no—
354Q8—31no—
355Q8—31no—
356Q8—61no—
357Q8#89311yescharacteristic
358D4—62no—
359D4—62no—
360D4—62no—
361D4—62no—
362D4—121no—
363D4—121no—
364D4—121no—
365D4—121no—
366D4—128no—
367D4—244no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—32no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—62no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—244no—
397C2×C4—244no—
398C2×C4C4×S412yes—
399C2×C4#98612yes—
400C8—62no—
401C8—62no—
402C8—62no—
403C8—121no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—41no—
411D3—84no—
412D3—162no—
413C22—122no—
414C22—122no—
415C22—122no—
416C22—122no—
417C22—122no—
418C22—122no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22C2×C4×S411yescharacteristic
428C22D4×S411yescharacteristic
429C22#591911yescharacteristic
430C22C2×#98611yescharacteristic
431C22#592311yescharacteristic
432C22#592611yescharacteristic
433C22#592611yescharacteristic
434C4—22no—
435C4—22no—
436C4—31no—
437C4—31no—
438C4—31no—
439C4—31no—
440C4—41no—
441C4—62no—
442C4—121no—
443C4—122no—
444C4—121no—
445C4—241no—
446C3—41noSylow
447C2—242no—
448C2—242no—
449C2#8530811yescharacteristic
450C2C4×#98611yescharacteristic
451C2#8535311yescharacteristic
452C2#8538211yescharacteristic
453C2#8538511yescharacteristic
454C2#8542111yescharacteristic
455C2#8542211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2351D3
prime radicalC2×C4⋊GL(2,𝔽3)C2
cosocleC22×SL(2,𝔽3)C23
hypercenter#95S4
hypocenterSL(2,𝔽3)C4×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
1. C2×#5394C2semidirect
3. C2×C4×GL(2,𝔽3)C2semidirect
4. #2351⋊C3C2semidirect
5. #73065C2semidirect
6. #73065C2semidirect
7. #73063C2semidirect
17. C4⋊GL(2,𝔽3)C4semidirect
24. #2351D3semidirect
42. C2×GL(2,𝔽3)C2×C4semidirect
43. C2×GL(2,𝔽3)C2×C4semidirect
49. C4×SL(2,𝔽3)C2×C4semidirect
51. #859D4semidirect
150. C4×Q8C4×D3semidirect
225. SL(2,𝔽3)C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#53942
C2×C4⋊GL(2,𝔽3)2
C2×C4×GL(2,𝔽3)2
#2351⋊C32
#730652
#730632
#230103
#46654