Finite Group Databasebeta

← #11780024

Subgroups — #11780024

Order 1152.

432 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 94 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
1Dic62C211yesmaximal, characteristic
2C3×#4851C211yesmaximal, characteristic
3#10590384C211yesmaximal, characteristic
4#10590288C211yesmaximal, characteristic
5#10589347C211yesmaximal, characteristic
6C3×#4484C211yesmaximal, characteristic
7#10589084C211yesmaximal, characteristic
8#71129—31nomaximal
9#68271—31nomaximal
10#64078C2211yescharacteristic
11C3×#869C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C3×C4×Dic6C2211yescharacteristic
14#63918—21no—
15#63918—21no—
16Dic3×Dic6—21no—
17Dic3×Dic6—21no—
18#63840—22no—
19C2×#3718—21no—
20C32×C4⋊C8C2211yescharacteristic
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#817—21no—
23C3×#816—21no—
24C3×C4×C3⋊C8C2211yescharacteristic
25#63567—22no—
26#63565—21no—
27Q8×Dic6—31no—
28Q8×Dic6—31no—
29C3×#475—31no—
30#5093—31no—
31#5090—31no—
32#5031—31no—
33#4851D311yescharacteristic
34#4681—31no—
35C3×Q8⋊C8—31no—
36#4535—31no—
37#4484D311yescharacteristic
38#4481—31no—
39#4481—31no—
40#4481—61no—
41#4456—31no—
42C2×#634D412yes—
43C2×C3×Dic6—21no—
44C2×C3×Dic6D411yescharacteristic
45C2×C3×Dic6D411yescharacteristic
46C2×#627—22no—
47C2×C32×C8—21no—
48C32×C42C2311yescharacteristic
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊C8)—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3D411yescharacteristic
55C3×C4⋊Dic3D411yescharacteristic
56C3×Dic3⋊C4—21no—
57C3×Dic3⋊C4—21no—
58C3×C4×Dic3—21no—
59C3×C4×Dic3—21no—
60C2×C3×C3⋊C8—21no—
61Dic3⋊Dic3—22no—
62Dic3⋊Dic3—22no—
63Dic32—22no—
64#3718—42no—
65#1645—91noSylow
66C3×C4⋊Q8—31no—
67C3×C4×Q8—31no—
68C3×C4×Q8—31no—
69Q8×Dic3—61no—
70Q8×Dic3—61no—
71C2×#216—61no—
72C2×#216—61no—
73#888—61no—
74#888—61no—
75#888—62no—
76#888—62no—
77#869—31no—
78#869—31no—
79#869—61no—
80#869—61no—
81#869—61no—
82#869C2×D311yescharacteristic
83C4×Dic6—61no—
84C4×Dic6—61no—
85C4×Dic6C2×D311yescharacteristic
86C4×Dic6C2×D311yescharacteristic
87C3×#106—61no—
88C3×C4⋊C8—21no—
89C3×C4⋊C8C2×D311yescharacteristic
90C3×C4⋊C8C2×D311yescharacteristic
91C3×#103—61no—
92C3×C4×C8—31no—
93#835—61no—
94#817—21no—
95#816—21no—
96#816—62no—
97#810—62no—
98#810—61no—
99C4⋊(C3⋊C8)—31no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—61no—
102C4×C3⋊C8C2×D311yescharacteristic
103C2×C32×C4C2×D411yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C3⋊Dic3—22no—
107#634—42no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—41no—
113C3×Dic6—21no—
114C3×Dic6—21no—
115C3×Dic6—41no—
116C3×Dic6—41no—
117C3×Dic6SD1612yes—
118C3×Dic6Q1612yescomplemented
119#627—82no—
120C32×C8—41no—
121C3⋊(C3⋊C8)—41no—
122C3×C3⋊C8—41no—
123Q82—91no—
124#487—91no—
125#475—31no—
126#462—91no—
127#333—91no—
128Q8⋊C8—31no—
129Q8⋊C8—91no—
130C2×C3×Q8—31no—
131C2×C3×Q8—31no—
132C2×C3×Q8—61no—
133C2×Dic6—21no—
134C2×Dic6—32no—
135C2×Dic6—32no—
136C2×Dic6—62no—
137C2×Dic6—62no—
138C2×Dic6—62no—
139C2×Dic6D1211yescharacteristic
140C2×Dic6#6711yescharacteristic
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—41no—
144C2×C3×C8—61no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—61no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—61no—
151C3×C42—21no—
152C3×C42—61no—
153C3×C42—61no—
154C3×C42C22×D311yescharacteristic
155C3×C42C22×D311yescharacteristic
156#216—122no—
157#216—122no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—62no—
161C4⋊Dic3—62no—
162C4⋊Dic3—61no—
163C4⋊Dic3—61no—
164C4⋊Dic3—61no—
165C4⋊Dic3—61no—
166C4⋊Dic3—121no—
167C4⋊Dic3—121no—
168C4⋊Dic3D1211yescharacteristic
169C4⋊Dic3#6711yescharacteristic
170Dic3⋊C4—21no—
171Dic3⋊C4—21no—
172Dic3⋊C4—62no—
173Dic3⋊C4—62no—
174C4×Dic3—21no—
175C4×Dic3—21no—
176C4×Dic3—62no—
177C4×Dic3—62no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—121no—
182C22×C32#12011yescharacteristic
183C32×C4—21no—
184C32×C4C2×SD1611yescharacteristic
185C32×C4C2×Q1611yescharacteristic
186C32×C4#13711yescharacteristic
187C32×C4#13711yescharacteristic
188C3⋊Dic3—42no—
189C3×Dic3—22no—
190C3×Dic3—22no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—81no—
194C2×Q16—181no—
195C4⋊Q8—31no—
196C4⋊Q8—91no—
197C4⋊Q8—181no—
198C4⋊Q8—181no—
199C4⋊Q8—182no—
200C4×Q8—31no—
201C4×Q8—31no—
202C4×Q8—181no—
203C4×Q8—181no—
204#106—61no—
205C4⋊C8—91no—
206C4⋊C8D3211yescharacteristic
207#103—61no—
208#103—181no—
209#103—182no—
210C4×C8—31no—
211C3×Q8—32no—
212C3×Q8—32no—
213C3×Q8—61no—
214C3×Q8—61no—
215C3×Q8—121no—
216C3×Q8—121no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—121no—
225C2×C3×C4C2×D1211yescharacteristic
226C2×C3×C4D3×D411yescharacteristic
227C2×C3×C4D3×D411yescharacteristic
228C2×C3×C4D3×D411yescharacteristic
229C2×C3×C4D3×D411yescharacteristic
230C2×C3×C4C2×#6711yescharacteristic
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—41no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—122no—
239Dic6—21no—
240Dic6—21no—
241Dic6—41no—
242Dic6—41no—
243Dic6—122no—
244Dic6—122no—
245Dic6—242no—
246Dic6—242no—
247Dic6—242no—
248Dic6#20412yes—
249Dic6#21612yescomplemented
250C3×C8—41no—
251C3×C8—41no—
252C3×C8—81no—
253C3×C8—121no—
254C3⋊C8—41no—
255C3⋊C8—121no—
256C3⋊C8—121no—
257C3⋊C8—241no—
258C2×C32#44811yescharacteristic
259C2×C32#45111yescharacteristic
260C2×C32#45511yescharacteristic
261C2×Q8—31no—
262C2×Q8—31no—
263C2×Q8—61no—
264C2×Q8—92no—
265C2×Q8—182no—
266Q16—362no—
267C2×C8—21no—
268C2×C8—61no—
269C2×C8—181no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—182no—
277C4⋊C4—182no—
278C4⋊C4—181no—
279C4⋊C4—181no—
280C42—61no—
281C42—61no—
282C42—182no—
283C42C2×D3211yescharacteristic
284C22×C3—21no—
285C22×C3#88211yescharacteristic
286C22×C3#93711yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—62no—
295C3×C4—62no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4—241no—
299C3×C4C2×#20411yescharacteristic
300C3×C4#90911yescharacteristic
301C3×C4D3×SD1611yescharacteristic
302C3×C4#91511yescharacteristic
303C3×C4#91511yescharacteristic
304C3×C4D3×Q1611yescharacteristic
305C3×C4C2×#21611yescharacteristic
306C3×C4#95111yescharacteristic
307Dic3—22no—
308Dic3—22no—
309Dic3—41no—
310Dic3—41no—
311Dic3—81no—
312Dic3—122no—
313Dic3—122no—
314Dic3—242no—
315C32#164511yescharacteristic, complemented, Sylow
316Q8—32no—
317Q8—32no—
318Q8—61no—
319Q8—61no—
320Q8—121no—
321Q8—121no—
322Q8—362no—
323Q8—722no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—121no—
329C2×C4—182no—
330C2×C4D3×D1211yescharacteristic
331C2×C4#380211yescharacteristic
332C2×C4D3×#6711yescharacteristic
333C8—41no—
334C8—121no—
335C8—361no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3#473811yescharacteristic
340C2×C3#480911yescharacteristic
341C2×C3#482511yescharacteristic
342C2×C3#505011yescharacteristic
343C2×C3#517011yescharacteristic
344C2×C3#520211yescharacteristic
345C22#6392211yescharacteristic
346C4—21no—
347C4—62no—
348C4—62no—
349C4—121no—
350C4—121no—
351C4—241no—
352C4—362no—
353C4D3×#20411yescharacteristic
354C4#6380011yescharacteristic
355C4#6392911yescharacteristic
356C4D3×#21611yescharacteristic
357C3—21no—
358C3#6827111yescharacteristic, complemented
359C3#7112911yescharacteristic, complemented
360C2#1059060411yescharacteristic
361C2#1059082211yescharacteristic
362C2#1059101111yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socle, prime radicalC22×C32#120
hypercenterC4⋊C8D32
hypocenterC32#1645
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
118. C3×Dic6Q16semidirect
249. Dic6#216semidirect
315. C32#1645semidirect
358. C3#68271semidirect
359. C3#71129semidirect

Maximal subgroups

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

subgroupindex
Dic622
C3×#48512
#105903842
#105902882
#105893472
C3×#44842
#105890842
#711293
#682713