Finite Group Databasebeta

← #10707447

Subgroups — #10707447

Order 768.

398 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#78314C211yesmaximal, characteristic, complemented
2C3×#1616C211yesmaximal, characteristic
3#71881C211yesmaximal, characteristic
4#71875C211yesmaximal, characteristic
5#67652C211yesmaximal, characteristic, complemented
6#67459C211yesmaximal, characteristic, complemented
7#67455C211yesmaximal, characteristic, complemented
8#12131—31nomaximal, Sylow
9C3×#537C2211yescharacteristic
10C2×#951—21no—
11C3×#481—21no—
12C3×#400C2211yescharacteristic
13C2×#837—21no—
14#5034—21no—
15#4986C2211yescharacteristic
16#4982—21no—
17#4973C2211yescharacteristic
18#4657—21no—
19#4653—21no—
20#4652—21no—
21#4647—21no—
22#4646—21no—
23C3×#327—21no—
24C3×#324C2211yescharacteristic
25#4537—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2809—31no—
29#2007—31no—
30#1616—31no—
31#1616D311yescharacteristic
32#1353—31no—
33#1353—31no—
34#1319—31no—
35C2×C3×#41—21no—
36C22×Dic6D411yescharacteristic
37C3×C22⋊Q8—41no—
38C3×#121—21no—
39C3×#121—21no—
40C3×#121—41no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43#951—42no—
44#951—42no—
45C2×#216—41no—
46C2×#214—41no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56C3×#106—41no—
57C3×#102—41no—
58C3×C22⋊C8—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62#837—42no—
63#832—41no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic
71#830—21no—
72#808—41no—
73C2×#137—61no—
74#537C2×D311yescharacteristic
75#481—21no—
76#481—61no—
77C2×#104—61no—
78#400—31no—
79#400C2×D311yescharacteristic
80#398—61no—
81#398—61no—
82#391—31no—
83#391—61no—
84#389—61no—
85#387—61no—
86#387—61no—
87#343—31no—
88#327—21no—
89#327—61no—
90#324—31no—
91#324C2×D311yescharacteristic
92C3×#41—41no—
93C3×#41—41no—
94C3×#41—41no—
95C3×#41—41no—
96C2×C3×Q8—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—41no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×Dic6—21no—
112C2×Dic6—41no—
113C2×Dic6—41no—
114C2×Dic6SD1612yes—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122#216—44no—
123#214—44no—
124C4⋊Dic3—42no—
125C4⋊Dic3—41no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—81no—
129Dic3⋊C4—81no—
130C4×Dic3—41no—
131C4×Dic3—81no—
132#208—22no—
133#208—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C2×#41—21no—
137C22×Q8—31no—
138#137—122no—
139#137—122no—
140C2×Q16—121no—
141C2×SD16—121no—
142C2×M16—61no—
143C22⋊Q8—41no—
144#121—21no—
145#121—21no—
146#121—41no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4#6711yescharacteristic
154C2×C4⋊C4C22×D311yescharacteristic
155C2×C42—31no—
156C2×C42—61no—
157#106—41no—
158#106—121no—
159#104—122no—
160#102—41no—
161#102—121no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164#95—21no—
165#95—21no—
166#95—31no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95#6711yescharacteristic
174C23×C3—41no—
175C23×C3#12011yescharacteristic
176C3×Q8—41no—
177C3×Q8—41no—
178C3×D4—41no—
179C3×D4—41no—
180C3×D4—81no—
181C3×D4—81no—
182C3×D4—81no—
183C3×D4—81no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4C2×SD1611yescharacteristic
200C2×C3×C4#13711yescharacteristic
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213Dic6—42no—
214Dic6—81no—
215C3×C8—81no—
216C3⋊C8—42no—
217C3⋊C8—81no—
218#41—41no—
219#41—41no—
220#41—41no—
221#41—41no—
222C2×Q8—21no—
223C2×Q8—32no—
224C2×Q8—61no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4C2×#6711yescharacteristic
241Q16—124no—
242SD16—124no—
243M16—62no—
244M16—121no—
245C2×C8—41no—
246C2×C8—121no—
247C2×C8—121no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—121no—
253C4⋊C4—122no—
254C4⋊C4—122no—
255C4⋊C4—122no—
256C4⋊C4—241no—
257C4⋊C4—241no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—121no—
261C42—241no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3#44811yescharacteristic
268C22×C3#45311yescharacteristic
269C22×C3B(4,𝔽2)11yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—81no—
277Dic3—42no—
278Dic3—42no—
279Dic3—81no—
280Dic3—81no—
281C23—41no—
282C23#93711yescharacteristic
283Q8—41no—
284Q8—41no—
285Q8—122no—
286Q8—241no—
287D4—41no—
288D4—41no—
289D4—81no—
290D4—81no—
291D4—81no—
292D4—81no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4D3×SD1611yescharacteristic
321C2×C4#91511yescharacteristic
322C8—81no—
323C8—122no—
324C8—241no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—161no—
328C2×C3#159611yescharacteristic
329C2×C3#220111yescharacteristic
330C2×C3#220411yescharacteristic
331C22—21no—
332C22—21no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22#503211yescharacteristic
337C22#517011yescharacteristic
338C22#519811yescharacteristic
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—122no—
347C4—122no—
348C4—241no—
349C4—241no—
350C3#1213111yescharacteristic, complemented, Sylow
351C2—21no—
352C2—21no—
353C2—161no—
354C2#7097611yescharacteristic
355C2#7098811yescharacteristic
356C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1616C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#537C22
hypercenter#1616D3
hypocenterC3#12131
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1616

Complements

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

NH ≅ G/Ntype
1. #78314C2semidirect
5. #67652C2semidirect
6. #67459C2semidirect
7. #67455C2semidirect
350. C3#12131semidirect

Maximal subgroups

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

subgroupindex
#783142
C3×#16162
#718812
#718752
#676522
#674592
#674552
#121313