Finite Group Databasebeta

← #11713237

Subgroups — #11713237

Order 768.

540 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 55 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×#5420C211yesmaximal, characteristic, complemented
3#73151C211yesmaximal, characteristic, complemented
4#73151C211yesmaximal, characteristic, complemented
5#73148C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#24047—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C23×SL(2,𝔽3)C2211yescharacteristic
11#548⋊C3—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5420—22no—
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#4952—41nomaximal
19C2×#548D311yescharacteristic
20C2×#482—31no—
21C2×#419—31no—
22#2485—31no—
23#2042—31no—
24#2042—31no—
25#2036—31no—
26#2035—31no—
27#1895—31no—
28#1870—31no—
29#1865—31no—
30#1863—31no—
31#1788—31no—
32#1788—31no—
33#1787—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3—21no—
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)D412yes—
40C22×SL(2,𝔽3)C2311yescharacteristic
41C2×C3×C22⋊C4—41no—
42C2×#217—41no—
43C2×#217—41no—
44C2×Dic3⋊C4—41no—
45C4×SL(2,𝔽3)—21no—
46#859—21no—
47#859—21no—
48#859—21no—
49#859—21no—
50C3⋊#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C22×#41C2×D311yescharacteristic
54C23×Q8C2×D311yescharacteristic
55C22×M16—31no—
56#548—22no—
57#548—62no—
58C2×#124—31no—
59C2×C22⋊Q8—31no—
60C2×#121—31no—
61C2×C4×Q8C2×D311yescharacteristic
62C2×C4×D4—31no—
63C2×#117—31no—
64#482—64no—
65C2×#107—31no—
66C2×#106—31no—
67#419—62no—
68#419—62no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×#102—31no—
80C2×#102—31no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8—31no—
83C2×C22⋊C8—31no—
84#397—61no—
85#394—61no—
86#389—31no—
87#389—31no—
88#388—61no—
89#387—61no—
90#380—31no—
91#380—31no—
92#380—31no—
93#340—31no—
94#340—31no—
95#336—31no—
96#336—31no—
97C24×C3—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104#41⋊C3—41no—
105C2×SL(2,𝔽3)—22no—
106C2×SL(2,𝔽3)C2×D411yescharacteristic
107C2×SL(2,𝔽3)#4111yescharacteristic
108C2×SL(2,𝔽3)#4111yescharacteristic
109C3×C22⋊C4—82no—
110#217—82no—
111#217—82no—
112Dic3⋊C4—82no—
113C2×#41—21no—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—61no—
117C2×#41—61no—
118C2×#41—61no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—31no—
122C22×Q8—32no—
123C22×Q8—61no—
124C22×Q8—61no—
125C22×Q8#6712yes—
126C22×Q8C22×D311yescharacteristic
127C22×D4—31no—
128C22×D4—31no—
129C23×C4—31no—
130C23×C4—31no—
131C2×M16—62no—
132C2×M16—121no—
133C2×M16—121no—
134C22×C8—31no—
135C22×C8—31no—
136C22×C8—31no—
137C22×C8—31no—
138#124—62no—
139#124—62no—
140C22⋊Q8—62no—
141C22⋊Q8—62no—
142#121—34no—
143#121—62no—
144C4×Q8—21no—
145C4×Q8—61no—
146C4×D4—62no—
147C4×D4—62no—
148#117—62no—
149#117—62no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4S411yescharacteristic
165C2×C42—31no—
166C2×C42—61no—
167#107—62no—
168#106—62no—
169#103—61no—
170#103—61no—
171#103—61no—
172#103—61no—
173#103—62no—
174#103—62no—
175#103—61no—
176#103—61no—
177#103—61no—
178#103—61no—
179#103—121no—
180#103—121no—
181#102—62no—
182#102—62no—
183#102—121no—
184#102—121no—
185C22⋊C8—62no—
186C22⋊C8—121no—
187C22⋊C8—121no—
188#95—61no—
189#95—61no—
190#95—61no—
191#95—61no—
192#95—61no—
193C23×C3—42no—
194C23×C3—41no—
195C23×C3—81no—
196C23×C3—81no—
197C23×C3—81no—
198C23×C3—81no—
199C23×C3—81no—
200C23×C3—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×Dic3—42no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219SL(2,𝔽3)#12311yescharacteristic
220C24—31no—
221C24C2×S411yescharacteristic
222#41—41no—
223#41—122no—
224#41—121no—
225#41—121no—
226#41—121no—
227C2×Q8—22no—
228C2×Q8—31no—
229C2×Q8—31no—
230C2×Q8—31no—
231C2×Q8—32no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—122no—
241C2×Q8#23711yescharacteristic
242C2×Q8#23711yescharacteristic
243C2×Q8C2×#6711yescharacteristic
244C2×D4—32no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—62no—
251C2×D4—121no—
252C2×D4—121no—
253C22×C4—31no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4C2×S411yescharacteristic
273C22×C4C2×S411yescharacteristic
274M16—124no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C4⋊C4—32no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—122no—
299C4⋊C4—241no—
300C22⋊C4—22no—
301C22⋊C4—62no—
302C22⋊C4—64no—
303C22⋊C4—122no—
304C22⋊C4—122no—
305C22⋊C4—241no—
306C22⋊C4—241no—
307C42—61no—
308C42—61no—
309C42—122no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—161no—
321C22×C3—161no—
322C22×C3—161no—
323C22×C3—161no—
324C3×C4—161no—
325C3×C4—161no—
326Dic3—82no—
327Dic3—161no—
328Dic3—161no—
329Dic3—161no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23GL(2,ℤ/4)12yes—
344C23C22×S411yescharacteristic
345Q8—31no—
346Q8—31no—
347Q8—31no—
348Q8—61no—
349Q8—62no—
350Q8—62no—
351Q8—121no—
352Q8—121no—
353Q8#93511yescharacteristic
354D4—62no—
355D4—62no—
356D4—121no—
357D4—121no—
358D4—124no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—32no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C8—121no—
398C8—121no—
399C8—121no—
400C8—121no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—162no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22#591411yescharacteristic
424C22#591411yescharacteristic
425C22#592311yescharacteristic
426C22#592311yescharacteristic
427C22#592611yescharacteristic
428C22#592711yescharacteristic
429C22C2×GL(2,ℤ/4)11yescharacteristic
430C4—31no—
431C4—31no—
432C4—31no—
433C4—31no—
434C4—41no—
435C4—41no—
436C4—62no—
437C4—121no—
438C4—241no—
439C4—241no—
440C4—241no—
441C4—241no—
442C3—41noSylow
443C2—42no—
444C2—241no—
445C2#8537811yescharacteristic
446C2#8537911yescharacteristic
447C2#8538711yescharacteristic
448C2#8538811yescharacteristic
449C2#8549111yescharacteristic
450C2#8552211yescharacteristic
451C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#548D3
prime radicalC2×#548⋊C3C2
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#123
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×#5420C2semidirect
3. #73151C2semidirect
4. #73151C2semidirect
5. #73148C2semidirect
6. #73062C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#548⋊C32
C2×#54202
#731512
#731482
#730622
#730592
#240473
#49524