Finite Group Databasebeta

← #11738027

Subgroups — #11738027

Order 896.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90362C211yesmaximal, characteristic
2Q8×Dic14C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic
4#89956C211yesmaximal, characteristic, complemented
5#89954C211yesmaximal, characteristic, complemented
6#89938C211yesmaximal, characteristic
7SD16×Dic7C211yesmaximal, characteristic
8#89827C211yesmaximal, characteristic
9#89802C211yesmaximal, characteristic, complemented
10#89623C211yesmaximal, characteristic, complemented
11#89600C211yesmaximal, characteristic, complemented
12#89564C211yesmaximal, characteristic, complemented
13#89563C211yesmaximal, characteristic, complemented
14#89537C211yesmaximal, characteristic
15#89533C211yesmaximal, characteristic
16C2×C7×Q16C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507C2211yescharacteristic
24C2×#1144C2211yescharacteristic
25#6499C2211yescharacteristic
26D4×Dic7C2211yescharacteristic
27#6451—21no—
28#6451—21no—
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6442—21no—
32#6442—21no—
33#6441—21no—
34#6440—21no—
35#6432—21no—
36#6432C2211yescharacteristic
37C4×Dic14—21no—
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6391C2211yescharacteristic
49#6391C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3369—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×Dic7—21no—
64C2×Dic14—21no—
65C2×Dic14—21no—
66C2×Dic14—21no—
67C2×Dic14D411yescharacteristic
68C2×Dic14C2311yescharacteristic
69C7×Q16—22no—
70C7×SD16—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C7×C22⋊C4—21no—
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82#1144—22no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7D411yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4D411yescharacteristic
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106Q82—71no—
107#556—71no—
108#489—71no—
109#489—71no—
110#487—71no—
111#476—71no—
112#475—71no—
113#475—71no—
114#464—71no—
115#464D711yescharacteristic
116#462—71no—
117#446—71no—
118#441—71no—
119#441—71no—
120C4×SD16—71no—
121C23×C7—21no—
122C7×Q8—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8#4112yes—
126C7×D4—21no—
127C7×D4—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—42no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143Dic14—22no—
144Dic14—21no—
145Dic14—21no—
146Dic14—81no—
147Dic14—81no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C2×Q16—71no—
154C2×Q16C2×D711yescharacteristic
155C2×SD16C2×D711yescharacteristic
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—71no—
160C4⋊Q8—141no—
161C4⋊Q8—141no—
162C4⋊Q8—141no—
163#126—141no—
164#126—141no—
165#124—71no—
166#124C2×D711yescharacteristic
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8C2×D711yescharacteristic
175C4×D4—71no—
176#117—141no—
177#106—71no—
178#106—71no—
179#106—71no—
180C4⋊C8—71no—
181C4⋊C8—71no—
182C4⋊C8C2×D711yescharacteristic
183#103—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
189#103C2×D711yescharacteristic
190#102—71no—
191#102—71no—
192#102C2×D711yescharacteristic
193#97—71no—
194#97—71no—
195C4×C8—71no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7C22×D411yescharacteristic
200C4×C7—22no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7C2×#4111yescharacteristic
206C4×C72−1+411yescharacteristic
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7#13712yes—
215C2×Q8—71no—
216C2×Q8—71no—
217C2×Q8—141no—
218C2×Q8—141no—
219C2×Q8—141no—
220C2×Q8C22×D711yescharacteristic
221C2×Q8C22×D711yescharacteristic
222C2×D4C22×D711yescharacteristic
223C22×C4—141no—
224Q16—22no—
225Q16—142no—
226SD16—22no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4C22×D711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C22⋊C4—141no—
253C42—21no—
254C42—71no—
255C42—71no—
256C42—141no—
257C42—141no—
258C42—141no—
259C42—141no—
260C42C22×D711yescharacteristic
261C2×C7—81no—
262C2×C7#54811yescharacteristic
263C2×C7C2×#13711yescharacteristic
264C2×C7#57711yescharacteristic
265C23—21no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269Q8—142no—
270Q8—141no—
271Q8—141no—
272Q8—561no—
273Q8—561no—
274Q8—561no—
275Q8#115912yes—
276D4—21no—
277D4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—71no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—71no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—282no—
290C2×C4D4×D711yescharacteristic
291C2×C4D4×D711yescharacteristic
292C2×C4C23×D711yescharacteristic
293C8—21no—
294C8—21no—
295C8—141no—
296C8—141no—
297C7#336911yescharacteristic, complemented, Sylow
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D711yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—72no—
308C4—141no—
309C4—141no—
310C4—141no—
311C4—281no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4C2×#115911yescharacteristic
316C4#655411yescharacteristic
317C2—81no—
318C2#9028911yescharacteristic
319C2#9046111yescharacteristic
320C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#464D7
hypocenterC7#3369
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. Q8×Dic14C2semidirect
4. #89956C2semidirect
5. #89954C2semidirect
9. #89802C2semidirect
10. #89623C2semidirect
11. #89600C2semidirect
12. #89564C2semidirect
13. #89563C2semidirect
297. C7#3369semidirect

Maximal subgroups

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

subgroupindex
#903622
Q8×Dic142
C7×#4642
#899562
#899542
#899382
SD16×Dic72
#898272
#898022
#896232
#896002
#895642
#895632
#895372
#895332
#33697