Finite Group Databasebeta

← #11713394

Subgroups — #11713394

Order 768.

627 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 59 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
2C2×#5426C211yesmaximal, characteristic, complemented
3C2×#5425C211yesmaximal, characteristic, complemented
4C2×#5419C211yesmaximal, characteristic, complemented
5#73148C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#24449—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C23×SL(2,𝔽3)C2211yescharacteristic
11C22×#226C2211yescharacteristic
12#548⋊C3—22no—
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5426—22no—
15#5425—22no—
16#5419—22no—
17C2×#859C2211yescharacteristic
18C2×#859C2211yescharacteristic
19C2×#859C2211yescharacteristic
20#4948—41nomaximal
21C2×#548D311yescharacteristic
22C2×#470—31no—
23C2×#452—31no—
24C2×#451—31no—
25C2×#417—31no—
26#2417—31no—
27#2042—31no—
28#2034—31no—
29#2015—31no—
30#2000—31no—
31#1878—31no—
32#1872—31no—
33#1870—31no—
34#1865—31no—
35#1787—31no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3D412yes—
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)—21no—
42C22×SL(2,𝔽3)C2311yescharacteristic
43C2×#226—21no—
44C2×#226—21no—
45C2×#226—21no—
46C2×C3×C22⋊C4—41no—
47C2×#217—41no—
48C2×#217—41no—
49C2×C4⋊Dic3—41no—
50C2×Dic3⋊C4—41no—
51C2×Dic3⋊C4—41no—
52C4×SL(2,𝔽3)—21no—
53#859—21no—
54#859—21no—
55#859D412yescomplemented
56C3⋊#95—41no—
57C22×#41C2×D311yescharacteristic
58C23×Q8C2×D311yescharacteristic
59C22×Q16—31no—
60C22×Q16—31no—
61C22×SD16—31no—
62C22×M16—31no—
63#548—22no—
64#548—62no—
65C2×#124—31no—
66C2×C22⋊Q8—31no—
67C2×C22⋊Q8—31no—
68C2×C22⋊Q8—31no—
69C2×#121—31no—
70C2×C4×Q8C2×D311yescharacteristic
71C2×C4×D4—31no—
72C22×C4⋊C4—31no—
73#470—64no—
74#452—62no—
75#452—62no—
76#451—62no—
77#451—62no—
78C2×#106—31no—
79#417—62no—
80#417—62no—
81C2×#103—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×#102—31no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94#398—61no—
95#396—61no—
96#391—31no—
97#389—31no—
98#386—61no—
99#380—31no—
100#380—61no—
101#340—31no—
102#340—31no—
103#336—31no—
104#336—31no—
105C24×C3—41no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×Dic3—41no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112#41⋊C3—22no—
113C2×SL(2,𝔽3)—41no—
114C2×SL(2,𝔽3)C2×D411yescharacteristic
115C2×SL(2,𝔽3)C2×D411yescharacteristic
116C2×SL(2,𝔽3)#4111yescharacteristic
117#226—41no—
118C3×C22⋊C4—82no—
119#217—82no—
120#217—82no—
121C4⋊Dic3—82no—
122Dic3⋊C4—82no—
123Dic3⋊C4—82no—
124C2×#41—21no—
125C2×#41—21no—
126C2×#41—32no—
127C2×#41—61no—
128C2×#41—61no—
129C2×#41#6712yes—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—31no—
134C22×Q8—31no—
135C22×Q8—61no—
136C22×Q8—61no—
137C22×Q8—61no—
138C22×Q8C22×D311yescharacteristic
139C22×D4—31no—
140C22×D4—31no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4—61no—
144C2×Q16—62no—
145C2×Q16—61no—
146C2×Q16—61no—
147C2×Q16—61no—
148C2×Q16—62no—
149C2×Q16—61no—
150C2×Q16—61no—
151C2×Q16—61no—
152C2×Q16—121no—
153C2×SD16—62no—
154C2×SD16—121no—
155C2×SD16—121no—
156C2×M16—62no—
157C2×M16—62no—
158C2×M16—121no—
159C22×C8—31no—
160C22×C8—31no—
161C22×C8—31no—
162C22×C8—31no—
163#124—62no—
164#124—62no—
165C22⋊Q8—34no—
166C22⋊Q8—62no—
167C22⋊Q8—62no—
168C22⋊Q8—62no—
169C22⋊Q8—62no—
170C22⋊Q8—121no—
171#121—62no—
172#121—62no—
173C4×Q8—21no—
174C4×Q8—61no—
175C4×D4—62no—
176C4×D4—62no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—121no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4S411yescharacteristic
194C2×C42—31no—
195#106—62no—
196#103—32no—
197#103—32no—
198#103—62no—
199#103—62no—
200#103—61no—
201#103—61no—
202#103—62no—
203#103—61no—
204#103—61no—
205#103—121no—
206#102—62no—
207#102—62no—
208#102—62no—
209C22⋊C8—62no—
210C22⋊C8—62no—
211C22⋊C8—121no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216C23×C3—42no—
217C23×C3—41no—
218C23×C3—81no—
219C23×C3—81no—
220C23×C3—81no—
221C23×C3—81no—
222C23×C3—81no—
223C23×C3—81no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237C2×Dic3—81no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242SL(2,𝔽3)#12111yescharacteristic
243C24—31no—
244C24C2×S411yescharacteristic
245#41—22no—
246#41—62no—
247#41—62no—
248#41—62no—
249#41—122no—
250C2×Q8—31no—
251C2×Q8—31no—
252C2×Q8—31no—
253C2×Q8—32no—
254C2×Q8—41no—
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—61no—
265C2×Q8—61no—
266C2×Q8—122no—
267C2×Q8—121no—
268C2×Q8—121no—
269C2×Q8—121no—
270C2×Q8D3×D411yescharacteristic
271C2×Q8#23711yescharacteristic
272C2×Q8C2×#6711yescharacteristic
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—121no—
282C22×C4—31no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—122no—
302C22×C4—122no—
303C22×C4—241no—
304C22×C4C2×S411yescharacteristic
305C22×C4C2×S411yescharacteristic
306Q16—124no—
307Q16—121no—
308Q16—121no—
309Q16—121no—
310Q16—121no—
311SD16—124no—
312M16—124no—
313C2×C8—32no—
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—61no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—62no—
334C4⋊C4—122no—
335C4⋊C4—122no—
336C4⋊C4—122no—
337C4⋊C4—241no—
338C22⋊C4—22no—
339C22⋊C4—62no—
340C22⋊C4—64no—
341C22⋊C4—122no—
342C22⋊C4—122no—
343C22⋊C4—241no—
344C42—61no—
345C42—61no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—161no—
357C22×C3—161no—
358C22×C3—161no—
359C22×C3—161no—
360C3×C4—82no—
361C3×C4—161no—
362Dic3—82no—
363Dic3—82no—
364Dic3—161no—
365Dic3—161no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—32no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23#98412yes—
380C23C22×S411yescharacteristic
381Q8—31no—
382Q8—31no—
383Q8—31no—
384Q8—61no—
385Q8—62no—
386Q8—62no—
387Q8—121no—
388Q8—121no—
389Q8—121no—
390Q8—121no—
391Q8—121no—
392Q8—121no—
393Q8#93911yescharacteristic
394D4—62no—
395D4—62no—
396D4—121no—
397D4—121no—
398D4—124no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
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×C4—121no—
436C2×C4—244no—
437C2×C4GL(2,ℤ/4)12yes—
438C8—62no—
439C8—121no—
440C8—121no—
441C8—121no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—162no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22—121no—
464C22D4×S411yescharacteristic
465C22#591411yescharacteristic
466C22C2×#98411yescharacteristic
467C22#592311yescharacteristic
468C22#592611yescharacteristic
469C22#592711yescharacteristic
470C22C2×GL(2,ℤ/4)11yescharacteristic
471C4—22no—
472C4—31no—
473C4—31no—
474C4—31no—
475C4—31no—
476C4—41no—
477C4—121no—
478C4—122no—
479C4—121no—
480C4—241no—
481C4—241no—
482C4—241no—
483C3—41noSylow
484C2—42no—
485C2—122no—
486C2#8537611yescharacteristic
487C2#8537711yescharacteristic
488C2#8537811yescharacteristic
489C2#8538411yescharacteristic
490C2#8549511yescharacteristic
491C2#8551711yescharacteristic
492C2#8553811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#548D3
prime radicalC2×#548⋊C3C2
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
2. C2×#5426C2semidirect
3. C2×#5425C2semidirect
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×#548⋊C32
C2×#54262
C2×#54252
C2×#54192
#731482
#730612
#730592
#244493
#49484