Finite Group Databasebeta

← #10695598

Subgroups — #10695598

Order 768.

574 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2583C211yesmaximal, characteristic
2#82727C211yesmaximal, characteristic
3#82176C211yesmaximal, characteristic
4#71689C211yesmaximal, characteristic
5#71552C211yesmaximal, characteristic
6#69442C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic
8#9945—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#398—21no—
13C3×#395—21no—
14C3×#389—22no—
15C3×#385—21no—
16C3×#385—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5225—21no—
19#5211—21no—
20#5211C2211yescharacteristic
21#5210—21no—
22#5207C2211yescharacteristic
23#5207C2211yescharacteristic
24#5045—22no—
25#5044—21no—
26#5043—21no—
27#4986—21no—
28#4981—21no—
29#4978—22no—
30#4964—21no—
31#4538—22no—
32#4528—22no—
33#4523—22no—
34#3095—31no—
35#2583D311yescharacteristic
36#2484—31no—
37#1826—31no—
38#1511—31no—
39#1463—31no—
40#1284—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42D412yes—
62C2×#217—21no—
63C2×#217—21no—
64#926—21no—
65#926—21no—
66#926—21no—
67#926—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208—21no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—21no—
77#835—22no—
78#835—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#830—21no—
87#830D412yes—
88#808—22no—
89#807—22no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#117—31no—
92C22×C4⋊C4C2×D311yescharacteristic
93#485—62no—
94#484—61no—
95#483—61no—
96#428—61no—
97#419—61no—
98#407—31no—
99#407—31no—
100#400—61no—
101#398—21no—
102#398—61no—
103#395—21no—
104#389—22no—
105#387—62no—
106#385—21no—
107#385—21no—
108#385—61no—
109#381—31no—
110#381—61no—
111C4×C22⋊C4C2×D311yescharacteristic
112#339—62no—
113#328—62no—
114#324—62no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—44no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—82no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—42no—
154#217—42no—
155#217—41no—
156#217—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—44no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—42no—
165#208—42no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C22×Q8#6711yescharacteristic
169C23×C4#6711yescharacteristic
170C23×C4#6711yescharacteristic
171C23×C4C22×D311yescharacteristic
172C2×M16—61no—
173C2×M16—61no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178#117—61no—
179#117—61no—
180#117—61no—
181#117—61no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—61no—
200C2×C42#6712yes—
201#107—62no—
202#106—62no—
203#103—62no—
204#103—62no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—32no—
213#95—62no—
214#95—62no—
215#95—62no—
216#95—61no—
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C3×Q8—42no—
224C3×Q8—81no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—81no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12711yescharacteristic
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24C2×#6711yescharacteristic
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4#23711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287M16—122no—
288M16—122no—
289C2×C8—62no—
290C2×C8—121no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—44no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—82no—
305C4⋊C4—124no—
306C22⋊C4—24no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—122no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—42no—
313C42—121no—
314C42—121no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3#39211yescharacteristic
324C22×C3#45612yes—
325C22×C3#45812yes—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—44no—
331C3×C4—42no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4—81no—
335Dic3—82no—
336Dic3—81no—
337Dic3—81no—
338C23—22no—
339C23—22no—
340C23—41no—
341C23#93711yescharacteristic
342C23#93911yescharacteristic
343C23#95311yescharacteristic
344Q8—42no—
345Q8—81no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4#93711yescharacteristic
378C2×C4#93811yescharacteristic
379C2×C4#93911yescharacteristic
380C2×C4#94011yescharacteristic
381C8—122no—
382C8—241no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—41no—
386C2×C3#200511yescharacteristic
387C2×C3#201011yescharacteristic
388C2×C3#202211yescharacteristic
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22#509512yes—
398C22#522311yescharacteristic
399C22#524312yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—42no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—242no—
410C4—241no—
411C4—241no—
412C3#994511yescharacteristic, complemented, Sylow
413C2—22no—
414C2—22no—
415C2—41no—
416C2#7171511yescharacteristic
417C2#7179511yescharacteristic
418C2#7184911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2583C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#2583D3
hypocenterC3#9945
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
412. C3#9945semidirect

Maximal subgroups

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

subgroupindex
C3×#25832
#827272
#821762
#716892
#715522
#694422
#676162
#99453