Finite Group Databasebeta

← #11869018

Subgroups — #11869018

Order 1152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#544C211yesmaximal, characteristic
2#10595987C211yesmaximal, characteristic
3#10592793C212yesmaximal, complemented
4#10592790C212yesmaximal, complemented
5#10592762C212yesmaximal, complemented
6#10592762C212yesmaximal, complemented
7#10592703C211yesmaximal, characteristic
8#10592698C211yesmaximal, characteristic
9#10592697C211yesmaximal, characteristic
10C2×#63637C211yesmaximal, characteristic
11#10592663C211yesmaximal, characteristic
12#83480—34nomaximal
13C22×#634C2211yescharacteristic
14C32×C4⋊Q8—21no—
15C32×C4⋊Q8C2212yes—
16C32×#125—21no—
17C32×#125C2212yes—
18C32×C22⋊Q8—22no—
19C32×#117C2211yescharacteristic
20C32×#117C2211yescharacteristic
21C2×C32×C4⋊C4C2211yescharacteristic
22#64135C2211yescharacteristic
23C2×#3748C2211yescharacteristic
24C22×C3⋊(C3⋊C8)C2211yescharacteristic
25#64096—21no—
26#64096—21no—
27#64086—22no—
28C4×#634C2212yes—
29C4×#634C2212yes—
30#63637C2212yes—
31#63637C2212yes—
32#63637C2212yes—
33#63637C2212yes—
34#63635C2211yescharacteristic
35#63635C2212yes—
36#63635C2211yescharacteristic
37#63634C2211yescharacteristic
38#63634C2212yes—
39#63634C2211yescharacteristic
40C4⋊(C3⋊(C3⋊C8))C2212yes—
41C4⋊(C3⋊(C3⋊C8))C2212yes—
42C3×#544D314yes—
43#5516—34no—
44#5094—38no—
45#5091—38no—
46#5063—38no—
47#5063—38no—
48#5004—34no—
49#4999—34no—
50#4998—34no—
51C2×#810—34no—
52#4964—34no—
53C2×C32×Q8—22no—
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C22×C3⋊Dic3—21no—
57C2×#634—21no—
58C2×#634—22no—
59C2×#634Q814yescomplemented
60C2×#634C2312yes—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—22no—
63C32×C4⋊C4—22no—
64C32×C4⋊C4—22no—
65C32×C4⋊C4D411yescharacteristic
66C32×C4⋊C4D411yescharacteristic
67C32×C4⋊C4C2312yes—
68C32×C4⋊C4C2311yescharacteristic
69C32×C4⋊C4C2311yescharacteristic
70C32×C22⋊C4—21no—
71C32×C22⋊C4D412yes—
72C32×C42—21no—
73C32×C42C2312yes—
74#3757—21no—
75C4⋊(C3⋊Dic3)C2311yescharacteristic
76C4⋊(C3⋊Dic3)C2311yescharacteristic
77#3750—22no—
78#3750—22no—
79C4×C3⋊Dic3—22no—
80C4×C3⋊Dic3—22no—
81C4×C3⋊Dic3—21no—
82#3748—22no—
83C2×C3⋊(C3⋊C8)—22no—
84C2×C3⋊(C3⋊C8)C2311yescharacteristic
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2312yes—
87#3082—91noSylow
88C22×Dic6—34no—
89C3×C4⋊Q8—24no—
90C3×C4⋊Q8C2×D318yes—
91C3×#125—24no—
92C3×#125C2×D318yes—
93C3×C22⋊Q8—28no—
94C3×#117C2×D314yes—
95C3×#117C2×D314yes—
96C2×C3×C4⋊C4C2×D314yes—
97#926—34no—
98C2×#208—34no—
99C22×C3⋊C8—34no—
100#887—64no—
101#887—64no—
102#877—68no—
103C4×Dic6—38no—
104C4×Dic6—38no—
105#810—38no—
106#810—38no—
107#810—38no—
108#810—38no—
109#808—38no—
110#808—34no—
111#808—34no—
112#807—38no—
113#807—34no—
114#807—34no—
115C4⋊(C3⋊C8)—38no—
116C4⋊(C3⋊C8)—38no—
117C23×C32C2×D411yescharacteristic
118C32×Q8—82no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—81no—
123C2×C32×C4C2×D412yes—
124C2×C32×C4C2×D412yes—
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×Q812yes—
127C2×C32×C4#4112yes—
128C2×C32×C4C2411yescharacteristic
129C2×C3⋊Dic3—24no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133#634—24no—
134#634—22no—
135#634C2×Q814yes—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—22no—
139#544C3⋊D311yescharacteristic
140#519—91no—
141#479—92no—
142#479—92no—
143#477—92no—
144#475—92no—
145#428—91no—
146#427—91no—
147#424—91no—
148#419—91no—
149C2×#103—91no—
150C2×C3×Q8—28no—
151C22×C3×C4—24no—
152C22×C3×C4C22×D314yes—
153C22×Dic3—64no—
154C2×Dic6—316no—
155C2×Dic6—38no—
156C2×Dic6—68no—
157C2×Dic6—64no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—28no—
160C3×C4⋊C4—28no—
161C3×C4⋊C4—28no—
162C3×C4⋊C4#6714yes—
163C3×C4⋊C4#6714yes—
164C3×C4⋊C4C22×D318yes—
165C3×C4⋊C4C22×D314yes—
166C3×C4⋊C4C22×D314yes—
167C3×C22⋊C4—24no—
168C3×C22⋊C4#6718yes—
169C3×C42—24no—
170C3×C42C22×D318yes—
171#217—64no—
172C4⋊Dic3—34no—
173C4⋊Dic3—34no—
174Dic3⋊C4—68no—
175Dic3⋊C4—68no—
176C4×Dic3—68no—
177C4×Dic3—68no—
178C4×Dic3—64no—
179#208—68no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8—38no—
182C2×C3⋊C8—34no—
183C2×C3⋊C8—68no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32C22⋊Q812yes—
187C22×C32C22×D411yescharacteristic
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—42no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4C22⋊Q812yes—
194C32×C4C22×Q811yescharacteristic
195C32×C4C2×#4111yescharacteristic
196C3⋊Dic3—24no—
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C22×Q8—91no—
200C2×M16—91no—
201C22×C8—91no—
202C4⋊Q8—21no—
203C4⋊Q8C2×C3⋊D312yes—
204#125—21no—
205#125C2×C3⋊D312yes—
206C22⋊Q8—22no—
207C4×Q8—92no—
208C4×Q8—92no—
209C4×Q8—181no—
210C4×Q8—181no—
211#117—91no—
212#117—182no—
213#117C2×C3⋊D311yescharacteristic
214#117C2×C3⋊D311yescharacteristic
215C2×C4⋊C4C2×C3⋊D311yescharacteristic
216#107—91no—
217#107—91no—
218#107—92no—
219#106—91no—
220#106—91no—
221#106—92no—
222C4⋊C8—92no—
223C4⋊C8—92no—
224#103—92no—
225#103—92no—
226#103—92no—
227#103—92no—
228C23×C3C2×#6714yes—
229C3×Q8—88no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—28no—
233C2×C3×C4—84no—
234C2×C3×C4D3×Q818yes—
235C2×C3×C4#23918yes—
236C2×C3×C4C2×#6718yes—
237C2×C3×C4C2×#6718yes—
238C2×C3×C4C2×#6714yes—
239C2×C3×C4C23×D314yes—
240C2×Dic3—616no—
241C2×Dic3—68no—
242C2×Dic3—64no—
243C2×Dic3—64no—
244Dic6—316no—
245Dic6—616no—
246Dic6—68no—
247C3⋊C8—64no—
248C3⋊C8—64no—
249C3⋊C8—68no—
250C2×C32—22no—
251C2×C32C2×C22⋊Q811yescharacteristic
252C2×C32#57711yescharacteristic
253C2×C32#57811yescharacteristic
254C2×Q8—22no—
255C2×Q8—92no—
256C2×Q8—94no—
257C2×Q8—182no—
258C2×Q8—181no—
259C22×C4—21no—
260C22×C4—181no—
261C22×C4C22×C3⋊D311yescharacteristic
262M16—182no—
263C2×C8—91no—
264C2×C8—92no—
265C2×C8—91no—
266C2×C8—182no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—182no—
274C4⋊C4—182no—
275C4⋊C4#63811yescharacteristic
276C4⋊C4#63811yescharacteristic
277C4⋊C4C22×C3⋊D312yes—
278C4⋊C4C22×C3⋊D311yescharacteristic
279C4⋊C4C22×C3⋊D311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—181no—
282C22⋊C4#63812yes—
283C42—21no—
284C42—181no—
285C42—182no—
286C42—182no—
287C42C22×C3⋊D312yes—
288C22×C3—24no—
289C22×C3—24no—
290C22×C3#94618yes—
291C22×C3C22×#6714yes—
292C3×C4—28no—
293C3×C4—28no—
294C3×C4—48no—
295C3×C4—44no—
296C3×C4—44no—
297C3×C4#94618yes—
298C3×C4C2×D3×Q814yes—
299C3×C4C2×#23914yes—
300Dic3—616no—
301Dic3—124no—
302Dic3—124no—
303C32#308211yescharacteristic, complemented, Sylow
304C23C2×#63811yescharacteristic
305Q8—82no—
306Q8—94no—
307Q8—182no—
308Q8—184no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—81no—
313C2×C4—182no—
314C2×C4—184no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4Q8×C3⋊D312yes—
318C2×C4#383212yes—
319C2×C4C2×#63812yes—
320C2×C4C2×#63812yes—
321C2×C4C2×#63811yescharacteristic
322C2×C4C23×C3⋊D311yescharacteristic
323C8—181no—
324C8—181no—
325C8—182no—
326C2×C3—28no—
327C2×C3C2×#94614yes—
328C2×C3#583614yes—
329C2×C3#583814yes—
330C22—21no—
331C22—21no—
332C22#6415512yes—
333C22C22×#63811yescharacteristic
334C4—22no—
335C4—22no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—184no—
340C4—361no—
341C4—361no—
342C4#6415512yes—
343C4C2×Q8×C3⋊D311yescharacteristic
344C4C2×#383211yescharacteristic
345C3#8348014yescomplemented
346C2—22no—
347C2C2×#6415511yescharacteristic
348C2#1059630711yescharacteristic
349C2#1059630911yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#544C2
socleC22×C32C22×D4
prime radicalC23×C32C2×D4
hypercenter#544C3⋊D3
hypocenterC32#3082
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #10592793C2semidirect
4. #10592790C2semidirect
5. #10592762C2semidirect
6. #10592762C2semidirect
59. C2×#634Q8semidirect
303. C32#3082semidirect
345. C3#83480semidirect

Maximal subgroups

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

subgroupindex
C32×#5442
#105959872
#105927932
#105927902
#105927622
#105927032
#105926982
#105926972
C2×#636372
#105926632
#834803