Finite Group Databasebeta

← #10691610

Subgroups — #10691610

Order 768.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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×#5043C211yesmaximal, characteristic
2#73878C211yesmaximal, characteristic
3C3×#1520C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic
5#69924C211yesmaximal, characteristic
6#67678C211yesmaximal, characteristic
7#67625C211yesmaximal, characteristic
8#9256—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#877—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×#835C412yescomplemented
15#5212C2211yescharacteristic
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5043—21no—
19#5043—21no—
20#5043—22no—
21Dic3×C4⋊C4—21no—
22#4972—21no—
23C2×#808C412yescomplemented
24#4943—21no—
25#4654—22no—
26C3×#328—22no—
27#4551—22no—
28#4526—21no—
29#4526—21no—
30C2×#483—31no—
31#2344—31no—
32#1996—31no—
33#1823—31no—
34#1520D311yescharacteristic
35#1289—31no—
36#1282—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C23×Dic3—21no—
40C22×C3×C8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217D411yescharacteristic
52C2×#217D411yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3C2×C412yes—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D412yes—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—22no—
63C2×C4×Dic3—21no—
64C22×C3⋊C8C2×C412yes—
65#877—42no—
66#877—42no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95D412yes—
70#835—22no—
71#835—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76#830—21no—
77#830—21no—
78#808—22no—
79#808—22no—
80C2×C22⋊Q8C2×D311yescharacteristic
81C2×#117—61no—
82C22×C4⋊C4—31no—
83#483—62no—
84#483—61no—
85#483—61no—
86C2×#106—32no—
87C2×#103—32no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8C2×D311yescharacteristic
90#389—61no—
91#387—62no—
92#381—61no—
93#380—31no—
94#380C2×D311yescharacteristic
95C4×C4⋊C4—61no—
96C4×C22⋊C4—31no—
97#342—61no—
98#342—61no—
99#340—62no—
100#328—22no—
101C24×C3C2×D411yescharacteristic
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—42no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—42no—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×Dic3—22no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C22×Dic3—41no—
126C2×C3×C8—42no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—42no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137#217—21no—
138#217—21no—
139#217—21no—
140#217—41no—
141#217—41no—
142#217SD1612yes—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—42no—
148C4⋊Dic3C22⋊C412yes—
149C4⋊Dic3C22⋊C412yes—
150Dic3⋊C4—44no—
151C4×Dic3—44no—
152C4×Dic3—42no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155C22×Q8C4×D311yescharacteristic
156C23×C4—61no—
157C23×C4C22×D311yescharacteristic
158C22×C8—21no—
159C22×C8—32no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164#117—122no—
165#117—122no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—62no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4C4×D311yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—61no—
183C2×C42—62no—
184C2×C42—61no—
185#106—62no—
186#106—62no—
187#103—62no—
188#103—62no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#95—62no—
194#95—62no—
195#95—61no—
196#95—61no—
197#95#6712yes—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3C4×D411yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3#12111yescharacteristic
203C23×C3C2×SD1611yescharacteristic
204C23×C3#13511yescharacteristic
205C23×C3AGL(1,ℤ/8)11yescharacteristic
206C23×C3#13711yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C3×C8—81no—
246C3⋊C8—42no—
247C24D3×D411yescharacteristic
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—42no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—121no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4C2×C4×D311yescharacteristic
267C22×C4#23511yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4#23711yescharacteristic
271C22×C4C2×#6711yescharacteristic
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—62no—
275C2×C8—122no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—32no—
279C4⋊C4—32no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—122no—
288C4⋊C4—124no—
289C22⋊C4—32no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C42—122no—
300C42—124no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3#38611yescharacteristic
312C22×C3C4×SD1611yescharacteristic
313C22×C3#44011yescharacteristic
314C22×C3#45011yescharacteristic
315C22×C3#45211yescharacteristic
316C22×C3#46111yescharacteristic
317C22×C3#46311yescharacteristic
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—82no—
322C3×C4—81no—
323C3×C4—81no—
324C3×C4#41012yes—
325Dic3—42no—
326Dic3—42no—
327Dic3—42no—
328Dic3—81no—
329Dic3—161no—
330C23—41no—
331C23—41no—
332C23#67⋊C411yescharacteristic
333C23#88411yescharacteristic
334C23D3×SD1611yescharacteristic
335C23#91411yescharacteristic
336C23#91811yescharacteristic
337C23#91911yescharacteristic
338C23#93711yescharacteristic
339Q8—42no—
340Q8—81no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4D3×C22⋊C411yescharacteristic
374C2×C4#88311yescharacteristic
375C2×C4C4×#6711yescharacteristic
376C2×C4#93811yescharacteristic
377C8—81no—
378C8—122no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3#187711yescharacteristic
386C2×C3#189911yescharacteristic
387C2×C3#190511yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22#478811yescharacteristic
399C22#481711yescharacteristic
400C22#481811yescharacteristic
401C22#481911yescharacteristic
402C22#495611yescharacteristic
403C22#516911yescharacteristic
404C22#518711yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—82no—
409C4—81no—
410C4—81no—
411C4—122no—
412C4—122no—
413C4—122no—
414C4—241no—
415C4—481no—
416C4#474012yes—
417C3#925611yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2#7002911yescharacteristic
425C2#7027611yescharacteristic
426C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9256
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
14. C2×#835C4semidirect
23. C2×#808C4semidirect
417. C3#9256semidirect

Maximal subgroups

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

subgroupindex
C2×#50432
#738782
C3×#15202
#717012
#699242
#676782
#676252
#92563