Finite Group Databasebeta

← #11712609

Subgroups — #11712609

Order 768.

1125 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1125 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×2−1+4⋊C3C211yesmaximal, characteristic
2C2×#5420C211yesmaximal, characteristic, complemented
3C22×#859C211yesmaximal, characteristic, complemented
4#73151C212yesmaximal, complemented
5#73148C212yesmaximal, complemented
6#20603—31nomaximal, Sylow
7C2×2−1+4⋊C3—22no—
8C2×2−1+4⋊C3—22no—
9C2×2−1+4⋊C3C414yescomplemented
10C22×#41⋊C3C2211yescharacteristic
11C23×SL(2,𝔽3)C2211yescharacteristic
12C23×SL(2,𝔽3)C2211yescharacteristic
13#5420—22no—
14C2×#859—22no—
15C2×#859C2212yes—
16C2×#859C2212yes—
17#5220—41nomaximal
18C22×2−1+4D311yescharacteristic
19C2×#417—31no—
20C2×#419—31no—
21C2×#416—31no—
22C22×#103—31no—
23C2×#408—31no—
24#2311—31no—
25#1788—32no—
26#1788—32no—
27#1787—32no—
28#1787—32no—
29C22×C3×D4—41no—
30C23×Dic3—41no—
312−1+4⋊C3—28no—
32C2×#41⋊C3—21no—
33C2×#41⋊C3—21no—
34C2×#41⋊C3C2×C412yescomplemented
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)—21no—
37C22×SL(2,𝔽3)C2×C412yes—
38C22×SL(2,𝔽3)C2×C412yescomplemented
39C22×SL(2,𝔽3)D412yes—
40C22×SL(2,𝔽3)D412yes—
41C22×SL(2,𝔽3)C2311yescharacteristic
42C2×#217—42no—
43C2×#217—41no—
44#859—22no—
45#859—22no—
46C3⋊#95—42no—
47C2×2−1+4—22no—
48C2×2−1+4—22no—
49C2×2−1+4—34no—
50C2×2−1+4—62no—
51C2×2−1+4—62no—
52C2×2−1+4Dic314yescomplemented
53C22×#41—31no—
54C22×#41—31no—
55C22×#41—31no—
56C22×#41C2×D311yescharacteristic
57C23×Q8—31no—
58C23×Q8C2×D311yescharacteristic
59C23×Q8C2×D311yescharacteristic
60C22×M16—31no—
61C23×C8—31no—
62C2×#117—31no—
63C22×C4⋊C4—31no—
64#419—62no—
65#419—62no—
66#417—64no—
67#416—64no—
68C2×#103—32no—
69C2×#103—32no—
70C2×#103—32no—
71C2×#103—32no—
72C2×#103—32no—
73C2×#103—32no—
74C2×#103—62no—
75C2×#103—62no—
76C2×#102—32no—
77C2×#102—32no—
78#408—64no—
79C2×C22⋊C8—32no—
80C2×C22⋊C8—32no—
81#391—61no—
82#390—61no—
83#386—61no—
84#386—61no—
85#380—32no—
86#380—32no—
87C24×C3—41no—
88C24×C3—41no—
89C2×C3×D4—44no—
90C2×C3×D4—82no—
91C2×C3×D4—82no—
92C22×C3×C4—41no—
93C22×Dic3—42no—
94C22×Dic3—42no—
95C22×Dic3—82no—
96C22×Dic3—84no—
97#41⋊C3—22no—
98C2×SL(2,𝔽3)—22no—
99C2×SL(2,𝔽3)C22⋊C414yescomplemented
100C2×SL(2,𝔽3)C22×C411yescharacteristic
101C2×SL(2,𝔽3)C2×D411yescharacteristic
102C2×SL(2,𝔽3)C2×D411yescharacteristic
103#217—82no—
104#217—84no—
1052−1+4—28no—
1062−1+4—68no—
1072−1+4—68no—
1082−1+4—68no—
1092−1+4—616no—
110C2×#41—21no—
111C2×#41—21no—
112C2×#41—34no—
113C2×#41—34no—
114C2×#41—32no—
115C2×#41—34no—
116C2×#41—34no—
117C2×#41—34no—
118C2×#41—34no—
119C2×#41—62no—
120C2×#41—62no—
121C2×#41—62no—
122C2×#41—62no—
123C2×#41—62no—
124C2×#41—62no—
125C2×#41—62no—
126C2×#41—62no—
127C2×#41—62no—
128C2×#41—62no—
129C2×#41—62no—
130C2×#41—62no—
131C2×#41—61no—
132C2×#41—61no—
133C2×#41C2×Dic312yescomplemented
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—31no—
137C22×Q8—31no—
138C22×Q8—31no—
139C22×Q8—32no—
140C22×Q8—32no—
141C22×Q8—32no—
142C22×Q8—32no—
143C22×Q8—34no—
144C22×Q8—62no—
145C22×Q8—62no—
146C22×Q8—62no—
147C22×Q8—62no—
148C22×Q8—62no—
149C22×Q8—61no—
150C22×Q8—61no—
151C22×Q8C2×Dic312yes—
152C22×Q8C2×Dic312yescomplemented
153C22×Q8#6712yes—
154C22×Q8#6712yes—
155C22×Q8C22×D311yescharacteristic
156C22×D4—31no—
157C22×D4—31no—
158C22×D4—31no—
159C22×D4S411yescharacteristic
160C23×C4—31no—
161C23×C4—31no—
162C23×C4—31no—
163C23×C4—61no—
164C23×C4—61no—
165C2×M16—62no—
166C2×M16—62no—
167C2×M16—62no—
168C22×C8—32no—
169C22×C8—32no—
170C22×C8—62no—
171C22×C8—62no—
172C22×C8—62no—
173#117—62no—
174#117—62no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—62no—
182C2×C22⋊C4—62no—
183C2×C42—61no—
184#103—64no—
185#103—62no—
186#103—64no—
187#103—64no—
188#103—64no—
189#103—62no—
190#103—62no—
191#103—62no—
192#102—64no—
193#102—64no—
194C22⋊C8—64no—
195C22⋊C8—64no—
196#95—62no—
197#95—62no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3—41no—
203C23×C3—82no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3—81no—
207C23×C3—81no—
208C23×C3—82no—
209C3×D4—88no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×Dic3—44no—
214C2×Dic3—82no—
215C2×Dic3—82no—
216C2×Dic3—82no—
217C2×Dic3—82no—
218C2×Dic3—82no—
219C2×Dic3—88no—
220SL(2,𝔽3)C2×C22⋊C411yescharacteristic
221C24—31no—
222C24C2×S411yescharacteristic
223C24C2×S411yescharacteristic
224#41—22no—
225#41—316no—
226#41—68no—
227#41—68no—
228#41—68no—
229#41—68no—
230#41—68no—
231#41—68no—
232#41—68no—
233#41—64no—
234#41—68no—
235#41—68no—
236#41—68no—
237#41—68no—
238#41—62no—
239#41—62no—
240#41—62no—
241C2×Q8—22no—
242C2×Q8—34no—
243C2×Q8—32no—
244C2×Q8—32no—
245C2×Q8—31no—
246C2×Q8—31no—
247C2×Q8—32no—
248C2×Q8—32no—
249C2×Q8—31no—
250C2×Q8—34no—
251C2×Q8—34no—
252C2×Q8—34no—
253C2×Q8—32no—
254C2×Q8—32no—
255C2×Q8—62no—
256C2×Q8—62no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—64no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—68no—
265C2×Q8—68no—
266C2×Q8—64no—
267C2×Q8—68no—
268C2×Q8—68no—
269C2×Q8—62no—
270C2×Q8—62no—
271C2×Q8—62no—
272C2×Q8#21714yescomplemented
273C2×Q8C22×Dic311yescharacteristic
274C2×Q8C2×#6711yescharacteristic
275C2×Q8C2×#6711yescharacteristic
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—32no—
279C2×D4—32no—
280C2×D4—32no—
281C2×D4—34no—
282C2×D4—34no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4SL(2,ℤ/4)14yes—
291C22×C4—32no—
292C22×C4—32no—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—64no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—64no—
314C22×C4—64no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—122no—
318C22×C4—242no—
319C22×C4C2×S411yescharacteristic
320M16—124no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—124no—
328C4⋊C4—62no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—122no—
333C22⋊C4—122no—
334C22⋊C4—242no—
335C22⋊C4—242no—
336C42—122no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—44no—
343C22×C3—82no—
344C22×C3—84no—
345C22×C3—82no—
346C22×C3—84no—
347C22×C3—82no—
348C22×C3—84no—
349C22×C3—161no—
350C22×C3—161no—
351C22×C3—162no—
352C3×C4—82no—
353Dic3—84no—
354Dic3—162no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—32no—
362C23—62no—
363C23—62no—
364C23—61no—
365C23—61no—
366C23C2×SL(2,ℤ/4)12yes—
367C23C2×SL(2,ℤ/4)12yes—
368C23GL(2,ℤ/4)12yes—
369C23GL(2,ℤ/4)12yes—
370C23C22×S411yescharacteristic
371Q8—34no—
372Q8—31no—
373Q8—31no—
374Q8—31no—
375Q8—34no—
376Q8—68no—
377Q8—61no—
378Q8—62no—
379Q8—62no—
380Q8—62no—
381Q8—62no—
382Q8—62no—
383Q8—62no—
384Q8C2×#21711yescharacteristic
385D4—28no—
386D4—34no—
387D4—68no—
388D4—62no—
389D4—62no—
390D4—62no—
391D4—68no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—31no—
395C2×C4—31no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—32no—
403C2×C4—34no—
404C2×C4—34no—
405C2×C4—34no—
406C2×C4—34no—
407C2×C4—32no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—616no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—244no—
426C2×C4C2×SL(2,ℤ/4)12yes—
427C8—122no—
428C8—122no—
429C2×C3—42no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—42no—
433C2×C3—41no—
434C2×C3—84no—
435C2×C3—162no—
436C22—24no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—24no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—64no—
446C22—62no—
447C22—121no—
448C22—121no—
449C22—122no—
450C22#543214yes—
451C22#592612yes—
452C22#592712yes—
453C22C22×SL(2,ℤ/4)11yescharacteristic
454C22C2×GL(2,ℤ/4)11yescharacteristic
455C22C2×GL(2,ℤ/4)11yescharacteristic
456C4—22no—
457C4—31no—
458C4—31no—
459C4—31no—
460C4—31no—
461C4—34no—
462C4—62no—
463C4—242no—
464C4—242no—
465C3—41noSylow
466C2—24no—
467C2—42no—
468C2—122no—
469C2#8553412yes—
470C2#8553511yescharacteristic
471C2#8553711yescharacteristic
472C2#8553912yes—
473C2C2×#543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC22×2−1+4D3
prime radicalC22×2−1+4⋊C3C2
cosocleC22×SL(2,𝔽3)C23
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)C2×C22⋊C4
LCSG ▷ C2×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. C22×#859C2semidirect
4. #73151C2semidirect
5. #73148C2semidirect
9. C2×2−1+4⋊C3C4semidirect
34. C2×#41⋊C3C2×C4semidirect
38. C22×SL(2,𝔽3)C2×C4semidirect
52. C2×2−1+4Dic3semidirect
99. C2×SL(2,𝔽3)C22⋊C4semidirect
133. C2×#41C2×Dic3semidirect
152. C22×Q8C2×Dic3semidirect
272. C2×Q8#217semidirect

Maximal subgroups

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

subgroupindex
C22×2−1+4⋊C32
C2×#54202
C22×#8592
#731512
#731482
#206033
#52204