Finite Group Databasebeta

← #10698059

Subgroups — #10698059

Order 768.

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

normal subgroups only

This group has 664 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
1C3×#3187C211yesmaximal, characteristic
2#82810C211yesmaximal, characteristic, complemented
3#82699C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic
6#69910C211yesmaximal, characteristic, complemented
7#67676C211yesmaximal, characteristic, complemented
8#10323—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#935—21no—
12C22×C4×Dic3C2211yescharacteristic
13C3×#483—21no—
14C3×#481—21no—
15C3×#465—22no—
16C3×#450—21no—
17C3×#448—21no—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5229—21no—
20#5221—21no—
21#5220—21no—
22#5212C2211yescharacteristic
23#5212C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#4978—22no—
26#4972—21no—
27#4972—21no—
28#4946—21no—
29#4944—22no—
30#4943—21no—
31#4550—22no—
32#4538—22no—
33#4537—22no—
34#3187D311yescharacteristic
35#2855—31no—
36#2854—31no—
37#1817—31no—
38#1520—31no—
39#1519—31no—
40#1289—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3D411yescharacteristic
46C23×Dic3D411yescharacteristic
47C2×C3×SD16—22no—
48C22×C3×C8D412yes—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—21no—
65C2×#217—21no—
66#935—42no—
67#935—42no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—21no—
73C2×C4×Dic3—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—21no—
78C22×C3⋊C8—21no—
79C3×#106—22no—
80C3×#103—22no—
81C3×#102—22no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95D412yes—
89C3⋊#95D412yes—
90#830—21no—
91#830—21no—
92C2×#123—61no—
93C2×C22⋊Q8C2×D311yescharacteristic
94C2×#121C2×D311yescharacteristic
95C22×C42—31no—
96#483—21no—
97#481—21no—
98#465—22no—
99#450—21no—
100#448—21no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103#391—61no—
104#390—61no—
105#389—61no—
106#389—61no—
107#388—62no—
108#387—62no—
109#386—61no—
110#385—61no—
111#381—61no—
112#380—31no—
113#380—31no—
114#336—62no—
115#328—62no—
116#327—62no—
117C24×C3—21no—
118C24×C3C2×D411yescharacteristic
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—42no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—42no—
153C22×Dic3SD1612yescomplemented
154C3×SD16—82no—
155C2×C3×C8—22no—
156C2×C3×C8—42no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167#217—44no—
168#217—44no—
169#217—41no—
170#217—41no—
171#217—41no—
172#217—41no—
173C4⋊Dic3—42no—
174C4⋊Dic3—42no—
175Dic3⋊C4—44no—
176Dic3⋊C4—44no—
177C4×Dic3—41no—
178C4×Dic3—41no—
179C4×Dic3—41no—
180C4×Dic3—41no—
181C2×C3⋊C8—42no—
182C2×C3⋊C8—41no—
183C22×Q8#6711yescharacteristic
184C22×D4—21no—
185C22×D4#6711yescharacteristic
186C23×C4—31no—
187C23×C4—31no—
188C23×C4C22×D311yescharacteristic
189C2×SD16—22no—
190C22×C8—61no—
191C22×C8#6712yes—
192#123—122no—
193#123—122no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C4⋊C4#6711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C42—61no—
216C2×C42—61no—
217C2×C42—61no—
218C2×C42—61no—
219C2×C42—61no—
220C2×C42—61no—
221#106—22no—
222#103—22no—
223#102—22no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—32no—
229#95—32no—
230#95—62no—
231#95—62no—
232#95—62no—
233#95—62no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3#12011yescharacteristic
242C23×C3#12011yescharacteristic
243C23×C3#12111yescharacteristic
244C23×C3C2×SD1611yescharacteristic
245C23×C3#13511yescharacteristic
246C23×C3AGL(1,ℤ/8)11yescharacteristic
247C23×C3#13711yescharacteristic
248C3×Q8—42no—
249C3×Q8—81no—
250C3×D4—42no—
251C3×D4—81no—
252C3×D4—81no—
253C3×D4—81no—
254C3×D4—81no—
255C3×D4—81no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12711yescharacteristic
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C3×C8—42no—
294C3⋊C8—81no—
295C24—21no—
296C24D3×D411yescharacteristic
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—32no—
313C22×C4—42no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4D3×D411yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4#23711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332SD16—82no—
333C2×C8—22no—
334C2×C8—42no—
335C2×C8—122no—
336C2×C8—121no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—122no—
344C4⋊C4—122no—
345C4⋊C4—124no—
346C4⋊C4—124no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—124no—
352C22⋊C4—124no—
353C22⋊C4—121no—
354C22⋊C4—121no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C42—121no—
358C42—121no—
359C42—121no—
360C42—121no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#39211yescharacteristic
377C22×C3#44811yescharacteristic
378C22×C3#44911yescharacteristic
379C22×C3#45011yescharacteristic
380C22×C3#45211yescharacteristic
381C22×C3#46511yescharacteristic
382C22×C3#46811yescharacteristic
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—81no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4#45812yes—
390Dic3—42no—
391Dic3—42no—
392Dic3—82no—
393Dic3—82no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23D3×SD1611yescharacteristic
402C23#91411yescharacteristic
403C23#91511yescharacteristic
404C23#91611yescharacteristic
405C23#93711yescharacteristic
406C23#93711yescharacteristic
407C23#93811yescharacteristic
408Q8—42no—
409Q8—81no—
410D4—42no—
411D4—81no—
412D4—81no—
413D4—81no—
414D4—81no—
415D4—81no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
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—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4#93911yescharacteristic
450C2×C4#93911yescharacteristic
451C2×C4#94011yescharacteristic
452C2×C4#95311yescharacteristic
453C8—42no—
454C8—241no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—161no—
462C2×C3#200011yescharacteristic
463C2×C3#201011yescharacteristic
464C2×C3#202211yescharacteristic
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22#516911yescharacteristic
481C22#517011yescharacteristic
482C22#517111yescharacteristic
483C22#517211yescharacteristic
484C22#517311yescharacteristic
485C22#517411yescharacteristic
486C22#522311yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—122no—
494C4—122no—
495C4—242no—
496C4—242no—
497C4#524312yes—
498C3#1032311yescharacteristic, complemented, Sylow
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—161no—
506C2#7184911yescharacteristic
507C2#7205411yescharacteristic
508C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#3187D3
hypocenterC3#10323
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #82810C2semidirect
4. #71690C2semidirect
6. #69910C2semidirect
7. #67676C2semidirect
153. C22×Dic3SD16semidirect
498. C3#10323semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
#828102
#826992
#716902
#716752
#699102
#676762
#103233