Finite Group Databasebeta

← #11713396

Subgroups — #11713396

Order 768.

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

normal subgroups only

This group has 655 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5420C211yesmaximal, characteristic, complemented
3C2×#5419C211yesmaximal, characteristic, complemented
4C2×#5419C211yesmaximal, characteristic, complemented
5#73148C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#24453—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C23×SL(2,𝔽3)C2211yescharacteristic
11C22×#226C2211yescharacteristic
12D4×SL(2,𝔽3)—22no—
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5420—22no—
15#5419—22no—
16#5419—22no—
17C2×#859C2211yescharacteristic
18C2×#859C2211yescharacteristic
19C2×#859C2211yescharacteristic
20#5222—41nomaximal
21C2×D4×Q8D311yescharacteristic
22C2×#470—31no—
23C2×#451—31no—
24C2×#451—31no—
25C2×#419—31no—
26#2415—31no—
27#2043—31no—
28#2035—31no—
29#2016—31no—
30#2001—31no—
31#1878—31no—
32#1873—31no—
33#1870—31no—
34#1865—31no—
35#1787—31no—
36C22×C3×D4—41no—
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)—21no—
42C22×SL(2,𝔽3)D412yes—
43C22×SL(2,𝔽3)C2311yescharacteristic
44C2×#226—21no—
45C2×#226—21no—
46C2×#226—21no—
47C2×#217—41no—
48C2×#217—41no—
49C2×#217—41no—
50C2×#217—41no—
51C2×Dic3⋊C4—41no—
52C4×SL(2,𝔽3)—21no—
53#859—21no—
54#859—21no—
55#859D412yescomplemented
56C3⋊#95—41no—
57C23×Q8C2×D311yescharacteristic
58C23×Q8C2×D311yescharacteristic
59C22×Q16—31no—
60C22×Q16—31no—
61C22×Q16—31no—
62C22×M16—31no—
63D4×Q8—22no—
64D4×Q8—62no—
65C2×C4⋊Q8—31no—
66C2×C22⋊Q8—31no—
67C2×C22⋊Q8—31no—
68C2×C22⋊Q8—31no—
69C2×C22⋊Q8—31no—
70C2×C4×Q8C2×D311yescharacteristic
71C2×C4×D4—31no—
72C2×#117—31no—
73#470—64no—
74#451—62no—
75#451—62no—
76#451—62no—
77#451—62no—
78C2×#106—31no—
79#419—62no—
80#419—62no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×#103—31no—
87C2×#103—31no—
88C2×#103—31no—
89C2×#103—31no—
90C2×#103—31no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94#397—61no—
95#394—61no—
96#391—31no—
97#389—31no—
98#389—61no—
99#388—61no—
100#380—31no—
101#340—31no—
102#340—31no—
103#336—31no—
104#336—31no—
105C24×C3—41no—
106C24×C3—41no—
107C2×C3×D4—82no—
108C2×C3×D4—82no—
109C2×C3×D4—82no—
110C22×C3×C4—41no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C2×SL(2,𝔽3)—22no—
116C2×SL(2,𝔽3)—41no—
117C2×SL(2,𝔽3)C2×D411yescharacteristic
118C2×SL(2,𝔽3)C2×D411yescharacteristic
119C2×SL(2,𝔽3)#4111yescharacteristic
120#226—41no—
121#217—82no—
122#217—82no—
123#217—82no—
124#217—82no—
125Dic3⋊C4—82no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—31no—
132C22×Q8—31no—
133C22×Q8—32no—
134C22×Q8—31no—
135C22×Q8—61no—
136C22×Q8—61no—
137C22×Q8—61no—
138C22×Q8—61no—
139C22×Q8—61no—
140C22×Q8#6712yes—
141C22×Q8C22×D311yescharacteristic
142C22×D4S411yescharacteristic
143C23×C4—31no—
144C23×C4—31no—
145C2×Q16—62no—
146C2×Q16—61no—
147C2×Q16—61no—
148C2×Q16—61no—
149C2×Q16—62no—
150C2×Q16—62no—
151C2×Q16—61no—
152C2×Q16—61no—
153C2×Q16—61no—
154C2×Q16—121no—
155C2×Q16—121no—
156C2×Q16—121no—
157C2×M16—62no—
158C2×M16—62no—
159C2×M16—121no—
160C22×C8—31no—
161C22×C8—31no—
162C22×C8—31no—
163C22×C8—31no—
164C4⋊Q8—62no—
165C4⋊Q8—62no—
166C22⋊Q8—34no—
167C22⋊Q8—62no—
168C22⋊Q8—62no—
169C22⋊Q8—62no—
170C22⋊Q8—62no—
171C22⋊Q8—62no—
172C22⋊Q8—62no—
173C22⋊Q8—121no—
174C4×Q8—21no—
175C4×Q8—61no—
176C4×D4—62no—
177C4×D4—62no—
178#117—62no—
179#117—121no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C42—31no—
196C2×C42—61no—
197#106—62no—
198#103—32no—
199#103—32no—
200#103—62no—
201#103—62no—
202#103—62no—
203#103—62no—
204#103—61no—
205#103—61no—
206#103—62no—
207#103—62no—
208#103—61no—
209#103—61no—
210#103—121no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—121no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—41no—
221C23×C3—81no—
222C23×C3—81no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3—81no—
230C23×C3—81no—
231C23×C3—81no—
232C23×C3—81no—
233C3×D4—88no—
234C2×C3×C4—42no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×Dic3—42no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249SL(2,𝔽3)#12111yescharacteristic
250C24C2×S411yescharacteristic
251C24C2×S411yescharacteristic
252C2×Q8—22no—
253C2×Q8—31no—
254C2×Q8—31no—
255C2×Q8—31no—
256C2×Q8—32no—
257C2×Q8—32no—
258C2×Q8—41no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—121no—
281C2×Q8—121no—
282C2×Q8—121no—
283C2×Q8D3×D411yescharacteristic
284C2×Q8#23711yescharacteristic
285C2×Q8C2×#6711yescharacteristic
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C22×C4—31no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4C2×S411yescharacteristic
310Q16—124no—
311Q16—121no—
312Q16—121no—
313Q16—121no—
314Q16—124no—
315Q16—121no—
316M16—124no—
317C2×C8—32no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—61no—
329C4⋊C4—32no—
330C4⋊C4—32no—
331C4⋊C4—62no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—64no—
335C4⋊C4—61no—
336C4⋊C4—61no—
337C4⋊C4—61no—
338C4⋊C4—61no—
339C4⋊C4—122no—
340C4⋊C4—122no—
341C4⋊C4—122no—
342C22⋊C4—62no—
343C22⋊C4—62no—
344C22⋊C4—122no—
345C22⋊C4—122no—
346C22⋊C4—122no—
347C22⋊C4—122no—
348C42—61no—
349C42—61no—
350C42—122no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—161no—
365C22×C3—161no—
366C22×C3—161no—
367C22×C3—161no—
368C22×C3—161no—
369C22×C3—161no—
370C22×C3—161no—
371C22×C3—161no—
372C3×C4—82no—
373Dic3—82no—
374Dic3—161no—
375Dic3—161no—
376Dic3—161no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23#98312yes—
390C23GL(2,ℤ/4)12yes—
391C23C22×S411yescharacteristic
392Q8—31no—
393Q8—31no—
394Q8—31no—
395Q8—61no—
396Q8—62no—
397Q8—62no—
398Q8—62no—
399Q8—62no—
400Q8—121no—
401Q8—121no—
402Q8—121no—
403Q8—121no—
404Q8—121no—
405Q8—121no—
406Q8—121no—
407Q8—121no—
408Q8#93911yescharacteristic
409D4—28no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—32no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—62no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
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—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4#98312yes—
448C8—62no—
449C8—121no—
450C8—121no—
451C8—121no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3—162no—
460C2×C3—162no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22D4×S411yescharacteristic
476C22#591411yescharacteristic
477C22C2×#98311yescharacteristic
478C22C2×#98311yescharacteristic
479C22#592711yescharacteristic
480C22#592711yescharacteristic
481C22C2×GL(2,ℤ/4)11yescharacteristic
482C4—22no—
483C4—31no—
484C4—31no—
485C4—31no—
486C4—31no—
487C4—62no—
488C4—121no—
489C4—122no—
490C4—121no—
491C4—241no—
492C4—241no—
493C4—241no—
494C3—41noSylow
495C2—42no—
496C2—42no—
497C2#8539111yescharacteristic
498C2#8539411yescharacteristic
499C2#8539511yescharacteristic
500C2#8539911yescharacteristic
501C2#8549511yescharacteristic
502C2#8550711yescharacteristic
503C2#8553511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×D4×Q8D3
prime radicalC2×D4×SL(2,𝔽3)C2
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5420C2semidirect
3. C2×#5419C2semidirect
4. C2×#5419C2semidirect
5. #73148C2semidirect
6. #73061C2semidirect
7. #73059C2semidirect
55. #859D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×SL(2,𝔽3)2
C2×#54202
C2×#54192
#731482
#730612
#730592
#244533
#52224