Finite Group Databasebeta

← #11713389

Subgroups — #11713389

Order 768.

718 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 65 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×#548⋊C3C211yesmaximal, characteristic, complemented
2C2×#5427C211yesmaximal, characteristic, complemented
3C2×#5421C211yesmaximal, characteristic, complemented
4C22×#859C211yesmaximal, characteristic, complemented
5#73151C211yesmaximal, characteristic, complemented
6#73065C211yesmaximal, characteristic, complemented
7#73062C211yesmaximal, characteristic, complemented
8#24439—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C23×SL(2,𝔽3)C2211yescharacteristic
11C22×GL(2,𝔽3)C2211yescharacteristic
12#548⋊C3—22no—
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5427—22no—
15#5421—22no—
16C2×#859—22no—
17C2×#859C2211yescharacteristic
18C2×#859C2211yescharacteristic
19C2×#859C2211yescharacteristic
20#4958—41nomaximal
21C2×#548D311yescharacteristic, complemented
22C2×#466—31no—
23C2×#449—31no—
24C2×#448—31no—
25C22×#103—31no—
26#2416—31no—
27#2042—31no—
28#2036—31no—
29#2012—31no—
30#2000—31no—
31#1880—31no—
32#1874—31no—
33#1869—31no—
34#1863—31no—
35#1788—31no—
36C22×#67—41no—
37C23×Dic3—41no—
38C2×#41⋊C3—21no—
39C2×#41⋊C3—21no—
40C2×#41⋊C3—21no—
41C22×SL(2,𝔽3)—21no—
42C22×SL(2,𝔽3)—21no—
43C22×SL(2,𝔽3)D412yescomplemented
44C22×SL(2,𝔽3)C2311yescharacteristic
45C2×GL(2,𝔽3)—21no—
46C2×GL(2,𝔽3)—21no—
47C2×GL(2,𝔽3)—21no—
48C2×C3×C22⋊C4—41no—
49C2×(C2×D3)⋊C4—41no—
50C2×(C2×D3)⋊C4—41no—
51C4×SL(2,𝔽3)—21no—
52#859—21no—
53#859—21no—
54#859D412yescomplemented
55C3⋊#95—41no—
56C3⋊#95—41no—
57C22×#41C2×D311yescharacteristic
58C23×Q8C2×D311yescharacteristic
59C22×SD16—31no—
60C22×SD16—31no—
61C22×D8—31no—
62C23×C8—31no—
63#548—22no—
64#548—62no—
65C2×#124—31no—
66C2×C22⋊Q8—31no—
67C2×#121—31no—
68C2×#121—31no—
69C2×#121—31no—
70C2×#121—61no—
71C2×C4×Q8C2×D311yescharacteristic
72C2×C4×D4—31no—
73C22×C4⋊C4—31no—
74#466—64no—
75#449—62no—
76#449—62no—
77#448—62no—
78#448—62no—
79C2×#107—31no—
80C2×#103—31no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×#103—62no—
87C2×#103—62no—
88C2×#102—31no—
89C2×#102—31no—
90C2×#102—31no—
91C2×#102—31no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8—31no—
95#396—61no—
96#394—61no—
97#390—31no—
98#389—31no—
99#386—61no—
100#380—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—44no—
110C2×#67—82no—
111C2×#67—82no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3—82no—
116C22×Dic3—82no—
117C22×Dic3—82no—
118#41⋊C3—41no—
119C2×SL(2,𝔽3)—22no—
120C2×SL(2,𝔽3)C2×D411yescharacteristic
121C2×SL(2,𝔽3)C2×D411yescharacteristic
122C2×SL(2,𝔽3)#4111yescharacteristic
123GL(2,𝔽3)—41no—
124C3×C22⋊C4—82no—
125(C2×D3)⋊C4—82no—
126(C2×D3)⋊C4—82no—
127C2×#41—21no—
128C2×#41—21no—
129C2×#41—21no—
130C2×#41—61no—
131C2×#41—61no—
132C2×#41—61no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—31no—
136C22×Q8—32no—
137C22×Q8—61no—
138C22×Q8—61no—
139C22×Q8#6712yes—
140C22×Q8C22×D311yescharacteristic
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145C22×D4—61no—
146C23×C4—31no—
147C23×C4—31no—
148C23×C4—61no—
149C2×SD16—61no—
150C2×SD16—61no—
151C2×SD16—62no—
152C2×SD16—61no—
153C2×SD16—61no—
154C2×SD16—61no—
155C2×SD16—61no—
156C2×SD16—121no—
157C2×SD16—121no—
158C2×D8—62no—
159C2×D8—62no—
160C2×D8—121no—
161C22×C8—31no—
162C22×C8—31no—
163C22×C8—31no—
164C22×C8—31no—
165C22×C8—62no—
166C22×C8—62no—
167C22×C8—121no—
168#124—62no—
169#124—62no—
170C22⋊Q8—62no—
171C22⋊Q8—62no—
172#121—34no—
173#121—62no—
174#121—62no—
175#121—62no—
176#121—62no—
177#121—121no—
178#121—124no—
179C4×Q8—21no—
180C4×Q8—61no—
181C4×D4—62no—
182C4×D4—62no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—121no—
192C2×C22⋊C4—31no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4S411yescharacteristic
200C2×C42—31no—
201#107—62no—
202#103—32no—
203#103—32no—
204#103—62no—
205#103—62no—
206#103—61no—
207#103—61no—
208#103—62no—
209#103—61no—
210#103—61no—
211#102—62no—
212#102—62no—
213#102—62no—
214#102—121no—
215C22⋊C8—62no—
216C22⋊C8—62no—
217C22⋊C8—121no—
218#95—61no—
219#95—61no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—81no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3—81no—
228C22×D3—42no—
229C22×D3—81no—
230C22×D3—81no—
231C22×D3—81no—
232C22×D3—81no—
233C22×D3—81no—
234C22×D3—81no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241#67—88no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251C2×Dic3—88no—
252SL(2,𝔽3)#12111yescharacteristic, complemented
253C24—31no—
254C24—61no—
255C24C2×S411yescharacteristic
256#41—41no—
257#41—122no—
258#41—121no—
259#41—121no—
260#41—121no—
261C2×Q8—22no—
262C2×Q8—31no—
263C2×Q8—31no—
264C2×Q8—31no—
265C2×Q8—32no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—61no—
271C2×Q8—62no—
272C2×Q8—62no—
273C2×Q8—62no—
274C2×Q8—122no—
275C2×Q8D3×D411yescharacteristic
276C2×Q8#23711yescharacteristic
277C2×Q8C2×#6711yescharacteristic
278C2×D4—32no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—122no—
294C2×D4—122no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—122no—
298C2×D4—241no—
299C22×C4—31no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4—241no—
320C22×C4C2×S411yescharacteristic
321C22×C4C2×S411yescharacteristic
322SD16—121no—
323SD16—121no—
324SD16—121no—
325SD16—124no—
326SD16—121no—
327D8—124no—
328C2×C8—32no—
329C2×C8—61no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8—61no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—124no—
341C4⋊C4—32no—
342C4⋊C4—32no—
343C4⋊C4—62no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—122no—
351C4⋊C4—122no—
352C22⋊C4—22no—
353C22⋊C4—62no—
354C22⋊C4—64no—
355C22⋊C4—122no—
356C22⋊C4—122no—
357C22⋊C4—122no—
358C22⋊C4—122no—
359C22⋊C4—241no—
360C42—61no—
361C42—61no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—44no—
370C22×C3—82no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—82no—
375C22×C3—82no—
376C2×D3—82no—
377C2×D3—82no—
378C2×D3—82no—
379C2×D3—161no—
380C2×D3—161no—
381C2×D3—161no—
382C2×D3—161no—
383C3×C4—161no—
384C3×C4—161no—
385Dic3—82no—
386Dic3—82no—
387Dic3—82no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—121no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23—121no—
404C23—121no—
405C23—121no—
406C23#98512yes—
407C23#98612yes—
408C23GL(2,ℤ/4)12yes—
409C23C22×S411yescharacteristic
410Q8—31no—
411Q8—31no—
412Q8—31no—
413Q8—61no—
414Q8—62no—
415Q8—62no—
416Q8—121no—
417Q8—121no—
418Q8#93911yescharacteristic
419D4—62no—
420D4—62no—
421D4—121no—
422D4—121no—
423D4—121no—
424D4—121no—
425D4—121no—
426D4—121no—
427D4—124no—
428D4—244no—
429D4—244no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—32no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—62no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—244no—
466C8—62no—
467C8—121no—
468C8—121no—
469C8—121no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—84no—
478D3—162no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—241no—
493C22—241no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22—241no—
499C22#542714yescomplemented
500C22D4×S411yescharacteristic
501C22#591411yescharacteristic
502C22C2×#98511yescharacteristic
503C22C2×#98611yescharacteristic
504C22#592611yescharacteristic
505C22#592711yescharacteristic
506C22C2×GL(2,ℤ/4)11yescharacteristic
507C4—31no—
508C4—31no—
509C4—31no—
510C4—31no—
511C4—41no—
512C4—41no—
513C4—62no—
514C4—121no—
515C4—122no—
516C4—241no—
517C4—241no—
518C3—41noSylow
519C2—24no—
520C2—241no—
521C2—481no—
522C2#8537411yescharacteristic
523C2#8538111yescharacteristic
524C2#8538211yescharacteristic
525C2#8538911yescharacteristic
526C2#8549511yescharacteristic
527C2C2×#542711yescharacteristic
528C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#548D3
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×#548⋊C3C2semidirect
2. C2×#5427C2semidirect
3. C2×#5421C2semidirect
4. C22×#859C2semidirect
5. #73151C2semidirect
6. #73065C2semidirect
7. #73062C2semidirect
21. C2×#548D3semidirect
43. C22×SL(2,𝔽3)D4semidirect
54. #859D4semidirect
252. SL(2,𝔽3)#121semidirect
499. C22#5427semidirect

Maximal subgroups

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

subgroupindex
C2×#548⋊C32
C2×#54272
C2×#54212
C22×#8592
#731512
#730652
#730622
#244393
#49584