Finite Group Databasebeta

← #11713240

Subgroups — #11713240

Order 768.

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

normal subgroups only

This group has 587 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5420C211yesmaximal, characteristic, complemented
3#73148C211yesmaximal, characteristic, complemented
4#73148C211yesmaximal, characteristic, complemented
5#73148C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#24053—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C23×SL(2,𝔽3)C2211yescharacteristic
11D4×SL(2,𝔽3)—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5420—22no—
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#5222—41nomaximal
19C2×D4×Q8D311yescharacteristic
20C2×#485—31no—
21C2×#419—31no—
22#2485—31no—
23#2043—31no—
24#2043—31no—
25#2036—31no—
26#2035—31no—
27#1894—31no—
28#1870—31no—
29#1865—31no—
30#1863—31no—
31#1787—31no—
32#1787—31no—
33#1787—31no—
34C22×C3×D4—41no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)—21no—
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)D412yes—
41C22×SL(2,𝔽3)C2311yescharacteristic
42C2×#217—41no—
43C2×#217—41no—
44C2×#217—41no—
45C2×#217—41no—
46C2×Dic3⋊C4—41no—
47C4×SL(2,𝔽3)—21no—
48#859—21no—
49#859—21no—
50#859—21no—
51#859—21no—
52C3⋊#95—41no—
53C23×Q8C2×D311yescharacteristic
54C23×Q8C2×D311yescharacteristic
55C22×M16—31no—
56D4×Q8—22no—
57D4×Q8—62no—
58C2×C4⋊Q8—31no—
59C2×C22⋊Q8—31no—
60C2×C22⋊Q8—31no—
61C2×C4×Q8C2×D311yescharacteristic
62C2×C4×D4—31no—
63C2×#117—31no—
64#485—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×#103—31no—
78C2×#103—31no—
79C2×#103—31no—
80C2×#103—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—
98C24×C3—41no—
99C2×C3×D4—82no—
100C2×C3×D4—82no—
101C2×C3×D4—82no—
102C22×C3×C4—41no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C2×SL(2,𝔽3)—22no—
108C2×SL(2,𝔽3)—41no—
109C2×SL(2,𝔽3)C2×D411yescharacteristic
110C2×SL(2,𝔽3)#4111yescharacteristic
111C2×SL(2,𝔽3)#4111yescharacteristic
112#217—82no—
113#217—82no—
114#217—82no—
115#217—82no—
116Dic3⋊C4—82no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—31no—
123C22×Q8—31no—
124C22×Q8—32no—
125C22×Q8—61no—
126C22×Q8—61no—
127C22×Q8—61no—
128C22×Q8—61no—
129C22×Q8—61no—
130C22×Q8#6712yes—
131C22×Q8C22×D311yescharacteristic
132C22×D4S411yescharacteristic
133C23×C4—31no—
134C23×C4—31no—
135C2×M16—62no—
136C2×M16—121no—
137C2×M16—121no—
138C22×C8—31no—
139C22×C8—31no—
140C22×C8—31no—
141C22×C8—31no—
142C4⋊Q8—62no—
143C4⋊Q8—62no—
144C22⋊Q8—34no—
145C22⋊Q8—62no—
146C22⋊Q8—62no—
147C22⋊Q8—62no—
148C4×Q8—21no—
149C4×Q8—61no—
150C4×D4—62no—
151C4×D4—62no—
152#117—62no—
153#117—62no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C42—31no—
170C2×C42—61no—
171#107—62no—
172#106—62no—
173#103—61no—
174#103—61no—
175#103—61no—
176#103—61no—
177#103—62no—
178#103—62no—
179#103—62no—
180#103—62no—
181#103—61no—
182#103—61no—
183#103—61no—
184#103—61no—
185#103—121no—
186#103—121no—
187#103—121no—
188#103—121no—
189C22⋊C8—62no—
190C22⋊C8—121no—
191C22⋊C8—121no—
192#95—61no—
193#95—61no—
194#95—61no—
195#95—61no—
196#95—61no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—41no—
200C23×C3—81no—
201C23×C3—81no—
202C23×C3—81no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3—81no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—81no—
212C3×D4—88no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×Dic3—42no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221C2×Dic3—81no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228SL(2,𝔽3)#12311yescharacteristic
229C24C2×S411yescharacteristic
230C24C2×S411yescharacteristic
231C2×Q8—22no—
232C2×Q8—31no—
233C2×Q8—31no—
234C2×Q8—31no—
235C2×Q8—32no—
236C2×Q8—32no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—62no—
249C2×Q8—62no—
250C2×Q8—62no—
251C2×Q8—122no—
252C2×Q8—122no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×Q8#23711yescharacteristic
257C2×Q8#23711yescharacteristic
258C2×Q8C2×#6711yescharacteristic
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C22×C4—31no—
263C22×C4—32no—
264C22×C4—32no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4C2×S411yescharacteristic
283M16—124no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C4⋊C4—32no—
297C4⋊C4—62no—
298C4⋊C4—61no—
299C4⋊C4—64no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—122no—
309C4⋊C4—241no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—122no—
313C22⋊C4—122no—
314C22⋊C4—241no—
315C22⋊C4—241no—
316C42—61no—
317C42—61no—
318C42—122no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—82no—
327C22×C3—82no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3—82no—
332C22×C3—161no—
333C22×C3—161no—
334C22×C3—161no—
335C22×C3—161no—
336C22×C3—161no—
337C22×C3—161no—
338C22×C3—161no—
339C22×C3—161no—
340C3×C4—82no—
341Dic3—82no—
342Dic3—161no—
343Dic3—161no—
344Dic3—161no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23#98312yes—
358C23GL(2,ℤ/4)12yes—
359C23C22×S411yescharacteristic
360Q8—31no—
361Q8—31no—
362Q8—31no—
363Q8—61no—
364Q8—62no—
365Q8—62no—
366Q8—62no—
367Q8—62no—
368Q8—121no—
369Q8—121no—
370Q8—121no—
371Q8—121no—
372Q8#93511yescharacteristic
373D4—28no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—32no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4#98312yes—
412C8—121no—
413C8—121no—
414C8—121no—
415C8—121no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—162no—
424C2×C3—162no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22#591411yescharacteristic
440C22#591411yescharacteristic
441C22C2×#98311yescharacteristic
442C22C2×#98311yescharacteristic
443C22#592611yescharacteristic
444C22#592711yescharacteristic
445C22C2×GL(2,ℤ/4)11yescharacteristic
446C4—22no—
447C4—31no—
448C4—31no—
449C4—31no—
450C4—31no—
451C4—62no—
452C4—121no—
453C4—121no—
454C4—241no—
455C4—241no—
456C4—241no—
457C4—241no—
458C3—41noSylow
459C2—42no—
460C2—42no—
461C2#8539411yescharacteristic
462C2#8539511yescharacteristic
463C2#8540011yescharacteristic
464C2#8540111yescharacteristic
465C2#8549111yescharacteristic
466C2#8550711yescharacteristic
467C2#8553911yescharacteristic
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)#123
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. #73148C2semidirect
4. #73148C2semidirect
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×D4×SL(2,𝔽3)2
C2×#54202
#731482
#730622
#730592
#240533
#52224