Finite Group Databasebeta

← #11713976

Subgroups — #11713976

Order 768.

692 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#5421C211yesmaximal, characteristic
3C2×#5419C211yesmaximal, characteristic, complemented
4C2×#5395C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73064C211yesmaximal, characteristic, complemented
7#73061C211yesmaximal, characteristic, complemented
8#32909—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11C22×#226C2211yescharacteristic
12C2×C4×SL(2,𝔽3)—22no—
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14C2×C4×SL(2,𝔽3)C2211yescharacteristic
15#5421—22no—
16#5419—22no—
17C2×#859C2211yescharacteristic
18C2×#859C2211yescharacteristic
19#5395—22no—
20#5213—41nomaximal
21C22×C4×Q8D311yescharacteristic, complemented
22C2×#464—31no—
23C2×#464—31no—
24C2×#451—31no—
25C2×#448—31no—
26#2850—31no—
27#1880—31no—
28#1880—31no—
29#1878—31no—
30#1878—31no—
31#1874—31no—
32#1872—31no—
33#1805—31no—
34#1805—31no—
35#1797—31no—
36C23×C3×C4—41no—
37C22×#67—41no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)D412yes—
41C22×SL(2,𝔽3)C2311yescharacteristic
42C2×GL(2,𝔽3)—21no—
43C2×GL(2,𝔽3)—21no—
44C2×GL(2,𝔽3)—21no—
45C2×#226—21no—
46C2×#226—21no—
47C2×#226—21no—
48C2×#217—41no—
49C2×(C2×D3)⋊C4—41no—
50C2×(C2×D3)⋊C4—41no—
51C4×SL(2,𝔽3)—21no—
52C4×SL(2,𝔽3)D412yescomplemented
53#859—21no—
54#859—21no—
55C3⋊#95—41no—
56C3⋊#95—41no—
57C23×Q8C2×D311yescharacteristic
58C22×Q16—31no—
59C22×Q16—31no—
60C22×SD16—31no—
61C22×SD16—31no—
62C2×#124—31no—
63C2×C22⋊Q8—31no—
64C2×#121—31no—
65C2×C4×Q8—22no—
66C2×C4×Q8—31no—
67C2×C4×Q8—31no—
68C2×C4×Q8—62no—
69C2×C4×Q8C2×D311yescharacteristic
70C2×C4×Q8C2×D311yescharacteristic
71C22×C4⋊C4—31no—
72C22×C42—31no—
73#464—62no—
74#464—64no—
75#464—62no—
76#451—62no—
77#451—62no—
78#448—62no—
79#448—62no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×#103—31no—
87C2×#103—31no—
88C2×#102—31no—
89C2×#102—31no—
90C2×C22⋊C8—31no—
91C2×C22⋊C8—31no—
92#391—31no—
93#390—31no—
94#388—61no—
95#388—61no—
96#386—61no—
97#386—61no—
98#382—31no—
99#340—31no—
100#340—31no—
101#340—31no—
102#340—31no—
103#336—31no—
104#336—31no—
105C24×C3—41no—
106C23×D3—41no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4—82no—
110C22×C3×C4—82no—
111C22×C3×C4—82no—
112C2×#67—44no—
113C2×#67—82no—
114C2×#67—82no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3—41no—
118C2×SL(2,𝔽3)—22no—
119C2×SL(2,𝔽3)C2×D411yescharacteristic
120C2×SL(2,𝔽3)C2×D411yescharacteristic
121C2×SL(2,𝔽3)#4111yescharacteristic
122GL(2,𝔽3)—41no—
123#226—41no—
124#217—82no—
125(C2×D3)⋊C4—82no—
126(C2×D3)⋊C4—82no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—31no—
130C22×Q8—32no—
131C22×Q8—31no—
132C22×Q8—61no—
133C22×Q8—61no—
134C22×Q8D1212yes—
135C22×Q8C22×D311yescharacteristic
136C22×D4—31no—
137C22×D4—61no—
138C23×C4—31no—
139C23×C4—31no—
140C23×C4S411yescharacteristic
141C2×Q16—61no—
142C2×Q16—62no—
143C2×Q16—62no—
144C2×Q16—61no—
145C2×Q16—61no—
146C2×Q16—62no—
147C2×Q16—61no—
148C2×Q16—61no—
149C2×Q16—61no—
150C2×SD16—61no—
151C2×SD16—62no—
152C2×SD16—62no—
153C2×SD16—61no—
154C2×SD16—61no—
155C2×SD16—62no—
156C2×SD16—61no—
157C2×SD16—61no—
158C2×SD16—61no—
159C22×C8—31no—
160C22×C8—31no—
161C22×C8—31no—
162C22×C8—31no—
163#124—62no—
164#124—62no—
165C22⋊Q8—62no—
166C22⋊Q8—62no—
167#121—62no—
168#121—62no—
169C4×Q8—21no—
170C4×Q8—32no—
171C4×Q8—34no—
172C4×Q8—62no—
173C4×Q8—62no—
174C4×Q8—61no—
175C4×Q8—62no—
176C4×Q8#6712yescomplemented
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C42—31no—
192C2×C42—31no—
193C2×C42—31no—
194C2×C42—31no—
195C2×C42—62no—
196C2×C42—62no—
197C4⋊C8—62no—
198C4⋊C8—62no—
199#103—62no—
200#103—62no—
201#103—62no—
202#103—61no—
203#103—62no—
204#103—61no—
205#103—61no—
206#103—61no—
207#102—62no—
208#102—62no—
209C22⋊C8—62no—
210C22⋊C8—62no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—81no—
220C23×C3—81no—
221C23×C3—81no—
222C23×C3—81no—
223C22×D3—42no—
224C22×D3—81no—
225C22×D3—81no—
226C22×D3—81no—
227C22×D3—81no—
228C22×D3—81no—
229C22×D3—81no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4—81no—
236C2×C3×C4—88no—
237#67—88no—
238C2×Dic3—42no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247SL(2,𝔽3)#12111yescharacteristic
248C24—61no—
249C24C2×S411yescharacteristic
250C2×Q8—22no—
251C2×Q8—32no—
252C2×Q8—31no—
253C2×Q8—31no—
254C2×Q8—31no—
255C2×Q8—32no—
256C2×Q8—32no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—64no—
267C2×Q8—62no—
268C2×Q8—62no—
269C2×Q8—62no—
270C2×Q8C2×D1211yescharacteristic
271C2×Q8#23511yescharacteristic
272C2×Q8C2×#6711yescharacteristic
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—122no—
280C2×D4—122no—
281C2×D4—122no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—31no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—31no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—121no—
301C22×C4C2×S411yescharacteristic
302C22×C4C2×S411yescharacteristic
303Q16—121no—
304Q16—121no—
305Q16—121no—
306Q16—124no—
307Q16—121no—
308SD16—121no—
309SD16—121no—
310SD16—121no—
311SD16—124no—
312SD16—121no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8—61no—
325C4⋊C4—32no—
326C4⋊C4—32no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—64no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—122no—
335C22⋊C4—122no—
336C22⋊C4—122no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C22⋊C4—122no—
340C22⋊C4—122no—
341C42—32no—
342C42—32no—
343C42—61no—
344C42—61no—
345C42—62no—
346C42—62no—
347C22×C3—44no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C2×D3—82no—
362C2×D3—82no—
363C2×D3—82no—
364C2×D3—161no—
365C2×D3—161no—
366C2×D3—161no—
367C2×D3—161no—
368C3×C4—82no—
369C3×C4—82no—
370Dic3—82no—
371Dic3—161no—
372Dic3—161no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—121no—
378C23—121no—
379C23—121no—
380C23—121no—
381C23—121no—
382C23—121no—
383C23—121no—
384C23C4⋊S412yes—
385C23#98312yes—
386C23#98512yes—
387C23C22×S411yescharacteristic
388Q8—31no—
389Q8—31no—
390Q8—31no—
391Q8—34no—
392Q8—61no—
393Q8—62no—
394Q8—62no—
395Q8—62no—
396Q8—121no—
397Q8—121no—
398Q8—121no—
399Q8—121no—
400Q8#93011yescharacteristic
401D4—121no—
402D4—121no—
403D4—121no—
404D4—121no—
405D4—244no—
406C2×C4—28no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—32no—
418C2×C4—32no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—124no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4#98312yes—
441C2×C4GL(2,ℤ/4)12yes—
442C8—121no—
443C8—121no—
444C8—121no—
445C8—121no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—84no—
454D3—162no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—241no—
462C22—241no—
463C22—241no—
464C22—241no—
465C22—241no—
466C22—241no—
467C22—241no—
468C22#539514yescomplemented
469C22C2×C4⋊S411yescharacteristic
470C22#591211yescharacteristic
471C22C2×#98311yescharacteristic
472C22C2×#98311yescharacteristic
473C22C2×#98511yescharacteristic
474C22#592311yescharacteristic
475C22C2×GL(2,ℤ/4)11yescharacteristic
476C4—22no—
477C4—22no—
478C4—31no—
479C4—31no—
480C4—31no—
481C4—31no—
482C4—62no—
483C4—62no—
484C4—121no—
485C4—241no—
486C4—241no—
487C4—241no—
488C3—41noSylow
489C2—24no—
490C2—481no—
491C2C2×#539511yescharacteristic
492C2#8532911yescharacteristic
493C2#8533011yescharacteristic
494C2#8533111yescharacteristic
495C2#8548611yescharacteristic
496C2#8550711yescharacteristic
497C2#8552111yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
prime radicalC2×#5421C2
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×SL(2,𝔽3)C2semidirect
3. C2×#5419C2semidirect
4. C2×#5395C2semidirect
5. #73065C2semidirect
6. #73064C2semidirect
7. #73061C2semidirect
21. C22×C4×Q8D3semidirect
52. C4×SL(2,𝔽3)D4semidirect
176. C4×Q8#67semidirect
468. C22#5395semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SL(2,𝔽3)2
C2×#54212
C2×#54192
C2×#53952
#730652
#730642
#730612
#329093
#52134