Finite Group Databasebeta

← #11713982

Subgroups — #11713982

Order 768.

635 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 61 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×#519⋊C3C211yesmaximal, characteristic, complemented
2C2×#5421C211yesmaximal, characteristic
3C2×#5419C211yesmaximal, characteristic, complemented
4C2×#5395C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7C4⋊#859C211yesmaximal, characteristic, complemented
8#32918—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11C22×#226C2211yescharacteristic
12#519⋊C3—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#4961—41nomaximal
21C2×#519D311yescharacteristic, complemented
22C2×#464—31no—
23C2×#464—31no—
24C2×#451—31no—
25C2×#448—31no—
26#2620—31no—
27#1880—31no—
28#1880—31no—
29#1878—31no—
30#1878—31no—
31#1875—31no—
32#1873—31no—
33#1803—31no—
34#1803—31no—
35#1798—31no—
36C22×#67—41no—
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)D412yes—
40C22×SL(2,𝔽3)C2311yescharacteristic
41C2×GL(2,𝔽3)—21no—
42C2×GL(2,𝔽3)—21no—
43C2×GL(2,𝔽3)—21no—
44C2×#226—21no—
45C2×#226—21no—
46C2×#226—21no—
47C2×C3×C22⋊C4—41no—
48C2×#217—41no—
49C2×(C2×D3)⋊C4—41no—
50C2×(C2×D3)⋊C4—41no—
51C2×C4⋊Dic3—41no—
52C4×SL(2,𝔽3)—21no—
53C4×SL(2,𝔽3)D412yescomplemented
54#859—21no—
55#859—21no—
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—
65#519—22no—
66#519—62no—
67C2×C4×Q8—31no—
68C2×C4×Q8—31no—
69C2×C4×Q8C2×D311yescharacteristic
70C2×C4×Q8C2×D311yescharacteristic
71C2×#117—31no—
72C2×#117—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#396—61no—
93#396—61no—
94#394—61no—
95#394—61no—
96#391—31no—
97#390—31no—
98#384—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—
109C2×#67—82no—
110C2×#67—82no—
111C2×#67—82no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C2×SL(2,𝔽3)—22no—
116C2×SL(2,𝔽3)C2×D411yescharacteristic
117C2×SL(2,𝔽3)C2×D411yescharacteristic
118C2×SL(2,𝔽3)#4111yescharacteristic
119GL(2,𝔽3)—41no—
120#226—41no—
121C3×C22⋊C4—82no—
122#217—82no—
123(C2×D3)⋊C4—82no—
124(C2×D3)⋊C4—82no—
125C4⋊Dic3—82no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—31no—
129C22×Q8—32no—
130C22×Q8—31no—
131C22×Q8—61no—
132C22×Q8—61no—
133C22×Q8D1212yes—
134C22×Q8C22×D311yescharacteristic
135C22×D4—31no—
136C22×D4—61no—
137C23×C4—31no—
138C2×Q16—61no—
139C2×Q16—62no—
140C2×Q16—62no—
141C2×Q16—61no—
142C2×Q16—61no—
143C2×Q16—62no—
144C2×Q16—61no—
145C2×Q16—61no—
146C2×Q16—61no—
147C2×SD16—61no—
148C2×SD16—62no—
149C2×SD16—62no—
150C2×SD16—61no—
151C2×SD16—61no—
152C2×SD16—62no—
153C2×SD16—61no—
154C2×SD16—61no—
155C2×SD16—61no—
156C22×C8—31no—
157C22×C8—31no—
158C22×C8—31no—
159C22×C8—31no—
160#124—62no—
161#124—62no—
162C22⋊Q8—62no—
163C22⋊Q8—62no—
164#121—62no—
165#121—62no—
166C4×Q8—21no—
167C4×Q8—32no—
168C4×Q8—34no—
169C4×Q8—62no—
170C4×Q8—62no—
171C4×Q8—62no—
172C4×Q8—61no—
173C4×Q8#6712yescomplemented
174#117—62no—
175#117—62no—
176#117—62no—
177#117—62no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4S411yescharacteristic
195C2×C42—31no—
196C2×C42—31no—
197C2×C42—61no—
198C4⋊C8—62no—
199C4⋊C8—62no—
200#103—62no—
201#103—62no—
202#103—62no—
203#103—61no—
204#103—62no—
205#103—61no—
206#103—61no—
207#103—61no—
208#102—62no—
209#102—62no—
210C22⋊C8—62no—
211C22⋊C8—62no—
212#95—61no—
213#95—61no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—81no—
217C23×C3—81no—
218C23×C3—81no—
219C23×C3—81no—
220C23×C3—81no—
221C23×C3—81no—
222C22×D3—42no—
223C22×D3—81no—
224C22×D3—81no—
225C22×D3—81no—
226C22×D3—81no—
227C22×D3—81no—
228C22×D3—81no—
229C2×C3×C4—42no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235#67—88no—
236C2×Dic3—42no—
237C2×Dic3—81no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245SL(2,𝔽3)#12111yescharacteristic
246C24—61no—
247C24C2×S411yescharacteristic
248C2×Q8—22no—
249C2×Q8—32no—
250C2×Q8—31no—
251C2×Q8—31no—
252C2×Q8—31no—
253C2×Q8—32no—
254C2×Q8—32no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—64no—
265C2×Q8—62no—
266C2×Q8—62no—
267C2×Q8—62no—
268C2×Q8C2×D1211yescharacteristic
269C2×Q8#23511yescharacteristic
270C2×Q8C2×#6711yescharacteristic
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—122no—
278C2×D4—241no—
279C2×D4—241no—
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4C2×S411yescharacteristic
294C22×C4C2×S411yescharacteristic
295Q16—121no—
296Q16—121no—
297Q16—121no—
298Q16—124no—
299Q16—121no—
300SD16—121no—
301SD16—121no—
302SD16—121no—
303SD16—124no—
304SD16—121no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—62no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—62no—
327C4⋊C4—122no—
328C4⋊C4—241no—
329C4⋊C4—241no—
330C22⋊C4—22no—
331C22⋊C4—62no—
332C22⋊C4—122no—
333C22⋊C4—122no—
334C22⋊C4—122no—
335C22⋊C4—122no—
336C22⋊C4—241no—
337C22⋊C4—241no—
338C42—32no—
339C42—32no—
340C42—61no—
341C42—61no—
342C42—64no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3—161no—
354C22×C3—161no—
355C22×C3—161no—
356C22×C3—161no—
357C2×D3—82no—
358C2×D3—82no—
359C2×D3—82no—
360C2×D3—161no—
361C2×D3—161no—
362C2×D3—161no—
363C2×D3—161no—
364C3×C4—82no—
365C3×C4—161no—
366Dic3—82no—
367Dic3—161no—
368Dic3—161no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—121no—
376C23—121no—
377C23—121no—
378C23—121no—
379C23—121no—
380C23—121no—
381C23—121no—
382C23C4⋊S412yes—
383C23C22×S411yescharacteristic
384Q8—31no—
385Q8—31no—
386Q8—31no—
387Q8—34no—
388Q8—61no—
389Q8—62no—
390Q8—62no—
391Q8—62no—
392Q8—121no—
393Q8—121no—
394Q8—121no—
395Q8—121no—
396Q8#93011yescharacteristic
397D4—121no—
398D4—121no—
399D4—121no—
400D4—121no—
401D4—244no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4GL(2,ℤ/4)12yes—
436C8—121no—
437C8—121no—
438C8—121no—
439C8—121no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—162no—
448D3—162no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—241no—
457C22—241no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22—241no—
463C22C2×C4⋊S411yescharacteristic
464C22#591211yescharacteristic
465C22#592611yescharacteristic
466C22#592611yescharacteristic
467C22#592711yescharacteristic
468C22#592711yescharacteristic
469C22C2×GL(2,ℤ/4)11yescharacteristic
470C4—22no—
471C4—31no—
472C4—31no—
473C4—31no—
474C4—31no—
475C4—41no—
476C4—62no—
477C4—62no—
478C4—121no—
479C4—241no—
480C4—241no—
481C4—241no—
482C3—41noSylow
483C2—42no—
484C2—481no—
485C2#8535511yescharacteristic
486C2#8535611yescharacteristic
487C2#8535711yescharacteristic
488C2#8535911yescharacteristic
489C2#8548611yescharacteristic
490C2#8553811yescharacteristic
491C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#519D3
prime radicalC2×#5421C2
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#519⋊C3C2semidirect
3. C2×#5419C2semidirect
4. C2×#5395C2semidirect
5. #73065C2semidirect
6. #73061C2semidirect
7. C4⋊#859C2semidirect
21. C2×#519D3semidirect
53. C4×SL(2,𝔽3)D4semidirect
173. C4×Q8#67semidirect

Maximal subgroups

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

subgroupindex
C2×#519⋊C32
C2×#54212
C2×#54192
C2×#53952
#730652
#730612
C4⋊#8592
#329183
#49614