Finite Group Databasebeta

← #10697289

Subgroups — #10697289

Order 768.

524 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3206C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic
3#83207C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic
5#71779C211yesmaximal, characteristic, complemented
6#70792C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#10197—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#492—21no—
13C3×#487—21no—
14C3×#461—22no—
15C3×#448—22no—
16C3×#431C2211yescharacteristic
17C2×#837—22no—
18#5094—21no—
19#5081—22no—
20#5080—21no—
21#5078—21no—
22#5075—21no—
23#5041—22no—
24#5019—21no—
25#5008—21no—
26#4964—22no—
27C4×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4529—22no—
31#4467—22no—
32#3246—31no—
33#3206D311yescharacteristic
34#3165—31no—
35#1987—31no—
36#1980—31no—
37#1503—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C2×C3×SD16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#214—22no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59#924—21no—
60#924—21no—
61C2×C4×Dic3—21no—
62C2×#208D412yes—
63C22×C3⋊C8D412yes—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×#103—24no—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73#837—44no—
74#832—22no—
75C3⋊#95—21no—
76#810—22no—
77#808—24no—
78#808—22no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87#501—62no—
88#500—61no—
89#498—61no—
90#492—21no—
91#487—21no—
92#476—61no—
93#475—61no—
94#465—62no—
95#461—22no—
96#461—61no—
97#460—61no—
98#448—22no—
99#431C2×D311yescharacteristic
100#428—62no—
101C2×C4⋊C8—31no—
102C2×#104—62no—
103C4×M16—31no—
104C4×C4⋊C4—31no—
105#340—62no—
106#328—62no—
107C3×#41—42no—
108C3×#41—42no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—82no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—41no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C3×SD16—84no—
126C2×C3×C8—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140#214—82no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C4×Dic3—42no—
151#208—22no—
152#208SD1614yescomplemented
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—42no—
156C2×#41—21no—
157C22×Q8—21no—
158C2×SD16—24no—
159C2×SD16—62no—
160C2×M16—32no—
161C22×C8—32no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8#6712yes—
168C4⋊Q8#6711yescharacteristic
169#127#6711yescharacteristic
170#125—61no—
171#124—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174#121—21no—
175#121—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—21no—
179C4×D4—21no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—61no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—64no—
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#104—124no—
193#103—24no—
194#103—62no—
195#102—62no—
196C22⋊C8#6712yes—
197#97—61no—
198C4×C8—21no—
199C4×C8—61no—
200#95—61no—
201C23×C3—41no—
202C23×C3#12711yescharacteristic
203C3×Q8—44no—
204C3×Q8—42no—
205C3×Q8—82no—
206C3×D4—42no—
207C3×D4—41no—
208C3×D4—41no—
209C3×D4—42no—
210C3×D4—81no—
211C3×D4—81no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—82no—
218C2×C3×C4—82no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4C2×SD1612yes—
226C2×C3×C4#13712yes—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232Dic6—41no—
233Dic6—41no—
234Dic6—81no—
235C3×C8—42no—
236C3⋊C8—24no—
237C3⋊C8—42no—
238#41—42no—
239#41—42no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—82no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—84no—
257SD16—242no—
258M16—34no—
259M16—62no—
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—122no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—62no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C42—121no—
282C42—122no—
283C42D3×D411yescharacteristic
284C42#23711yescharacteristic
285C42C2×#6711yescharacteristic
286C42C2×#6711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3#39211yescharacteristic
293C22×C3#49211yescharacteristic
294C22×C3#49711yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—82no—
299C3×C4—81no—
300C3×C4#44812yes—
301C3×C4#45511yescharacteristic
302C3×C4#45611yescharacteristic
303C3×C4#46112yes—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307C23—41no—
308C23#94011yescharacteristic
309Q8—44no—
310Q8—42no—
311Q8—82no—
312Q8—121no—
313Q8—121no—
314Q8—241no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—241no—
332C2×C4—241no—
333C2×C4D3×SD1612yes—
334C2×C4#91512yes—
335C2×C4#93711yescharacteristic
336C2×C4#93711yescharacteristic
337C2×C4#93811yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4#93911yescharacteristic
340C2×C4#95311yescharacteristic
341C8—42no—
342C8—64no—
343C8—122no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—161no—
347C2×C3#200711yescharacteristic
348C2×C3#200811yescharacteristic
349C2×C3#201311yescharacteristic
350C22—21no—
351C22—21no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#516711yescharacteristic
356C22#517511yescharacteristic
357C22#522311yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—82no—
363C4—242no—
364C4—241no—
365C4—241no—
366C4#516212yes—
367C4#517012yes—
368C4#520211yescharacteristic
369C4#520511yescharacteristic
370C3#1019711yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—161no—
374C2#7185811yescharacteristic
375C2#7205711yescharacteristic
376C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3206C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3206D3
hypocenterC3#10197
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
3. #83207C2semidirect
5. #71779C2semidirect
6. #70792C2semidirect
7. #67744C2semidirect
152. #208SD16semidirect
370. C3#10197semidirect

Maximal subgroups

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

subgroupindex
C3×#32062
#833202
#832072
#718182
#717792
#707922
#677442
#101973