Finite Group Databasebeta

← #32006934

Subgroups — #32006934

Order 18176.

1302 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986052C211yesmaximal, characteristic
2#22590747C211yesmaximal, characteristic
3#22589302C214yesmaximal, complemented
4#22589293C214yesmaximal, complemented
5#22588426C214yesmaximal
6C22×C4⋊Q8×C71C211yesmaximal, characteristic
7C2×#14556087—21no—
8#17986052—24no—
9C2×C4⋊(C71⋊C8)C2212yes—
10C2×C22⋊(C71⋊C8)C2212yes—
11C2×C4×C71⋊C8C2212yes—
12#17985869C2212yes—
13#17985865—21no—
14#17985859—21no—
15#17985859C2212yes—
16C4×C4⋊Dic71—21no—
17C4×C4⋊Dic71C2212yes—
18#17985761C22116yes—
19C23×Q8×C71—21no—
20C2×C4⋊Q8×C71—21no—
21C2×C4⋊Q8×C71—21no—
22C2×C4⋊Q8×C71—24no—
23C2×C4⋊Q8×C71C418yescomplemented
24C2×C4⋊Q8×C71C2212yes—
25C2×C4⋊Q8×C71C2212yes—
26C22×C4⋊C4×C71—21no—
27C22×C4⋊C4×C71C2212yes—
28C22×C42×C71C2211yescharacteristic
29C22×C71⋊C8C2314yes—
30#14556087—44no—
31C2×C4⋊Dic71—22no—
32C2×C4⋊Dic71—22no—
33C2×C4⋊Dic71C2314yes—
34C2×C22⋊Dic71—22no—
35C2×C4×Dic71—22no—
36C4⋊(C71⋊C8)—24no—
37C22⋊(C71⋊C8)—24no—
38C4×C71⋊C8—24no—
39C71⋊#95—22no—
40C71⋊#95—22no—
41C22×Q8×C71—28no—
42C22×Q8×C71—22no—
43C22×Q8×C71—22no—
44C22×Q8×C71—44no—
45C22×Q8×C71—44no—
46C23×C4×C71—22no—
47C23×C4×C71D411yescharacteristic
48C23×C4×C71D411yescharacteristic
49C23×C4×C71C2311yescharacteristic
50C4⋊Q8×C71—28no—
51C4⋊Q8×C71—28no—
52C4⋊Q8×C71—24no—
53C4⋊Q8×C71—24no—
54C4⋊Q8×C71C2×C418yes—
55C4⋊Q8×C71C2×C418yes—
56C2×C4⋊C4×C71—28no—
57C2×C4⋊C4×C71—22no—
58C2×C4⋊C4×C71—22no—
59C2×C4⋊C4×C71—22no—
60C2×C4⋊C4×C71—24no—
61C2×C4⋊C4×C71C2×C418yes—
62C2×C4⋊C4×C71C2314yes—
63C2×C42×C71C2×C414yes—
64C2×C42×C71D411yescharacteristic
65C2×C42×C71D411yescharacteristic
66C2×C42×C71D414yes—
67C2×C42×C71C2311yescharacteristic
68C2×C42×C71C2311yescharacteristic
69C22×Dic71—24no—
70C2×C71⋊C8—24no—
71C2×C71⋊C8—28no—
72C4⋊Dic71—28no—
73C4⋊Dic71—44no—
74C4⋊Dic71—44no—
75C22⋊Dic71—44no—
76C4×Dic71—44no—
77C24×C71C2×D411yescharacteristic
78C2×Q8×C71—28no—
79C2×Q8×C71—28no—
80C2×Q8×C71—432no—
81C2×Q8×C71—44no—
82C2×Q8×C71—44no—
83C2×Q8×C71—44no—
84C2×Q8×C71—44no—
85C22×C4×C71—24no—
86C22×C4×C71—24no—
87C22×C4×C71—28no—
88C22×C4×C71—24no—
89C22×C4×C71—44no—
90C22×C4×C71—44no—
91C22×C4×C71C22⋊C414yes—
92C22×C4×C71C22⋊C414yes—
93C22×C4×C71D814yes—
94C22×C4×C71SD1614yes—
95C22×C4×C71C22×C412yes—
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71C2×D411yescharacteristic
98C22×C4×C71C2×D412yes—
99C22×C4×C71C2×D411yescharacteristic
100C22×C4×C71C2×D411yescharacteristic
101C22×C4×C71C2×D411yescharacteristic
102C22×C4×C71C2411yescharacteristic
103C4⋊C4×C71—28no—
104C4⋊C4×C71—28no—
105C4⋊C4×C71—28no—
106C4⋊C4×C71—24no—
107C4⋊C4×C71C22×C418yes—
108C42×C71C22⋊C414yes—
109C42×C71C22⋊C414yes—
110C42×C71C22×C412yes—
111C42×C71C22×C412yes—
112C42×C71C2×D412yes—
113C42×C71C2×D412yes—
114C23×C71—22no—
115C23×C71—22no—
116C23×C71C2×C22⋊C412yes—
117C23×C71C2×D812yes—
118C23×C71C2×SD1612yes—
119C23×C71C22×D411yescharacteristic
120Q8×C71—416no—
121Q8×C71—416no—
122C2×C4×C71—22no—
123C2×C4×C71—22no—
124C2×C4×C71—24no—
125C2×C4×C71—24no—
126C2×C4×C71—28no—
127C2×C4×C71—28no—
128C2×C4×C71—44no—
129C2×C4×C71—44no—
130C2×C4×C71—416no—
131C2×C4×C71#102116yes—
132C2×C4×C71C2×C22⋊C412yes—
133C2×C4×C71C2×C22⋊C412yes—
134C2×C4×C71C2×C22⋊C412yes—
135C2×C4×C71C2×C22⋊C412yes—
136C2×C4×C71C2×C22⋊C412yes—
137C2×C4×C71C2×D812yes—
138C2×C4×C71C2×D812yes—
139C2×C4×C71C2×SD1612yes—
140C2×C4×C71C2×SD1612yes—
141C2×C4×C71#13714yes—
142C2×C4×C71C23×C411yescharacteristic
143C2×C4×C71C22×D411yescharacteristic
144C2×C4×C712−1+414yes—
145C2×Dic71—44no—
146C2×Dic71—44no—
147C2×Dic71—44no—
148C71⋊C8—44no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—24no—
152C22×C71—24no—
153C22×C71C2×#10214yes—
154C22×C71C22×C22⋊C411yescharacteristic
155C22×C71#53612yes—
156C22×C71C22×D811yescharacteristic
157C22×C71C22×SD1611yescharacteristic
158C22×C71C2×#13712yes—
159C4×C71—22no—
160C4×C71—22no—
161C4×C71—48no—
162C4×C71C2×#10214yes—
163C4×C71C2×#10214yes—
164C4×C71#41914yes—
165C4×C71#51912yes—
166C4×C71#53612yes—
167Dic71—84no—
168#21833—711nomaximal, Sylow
169C2×C71—24no—
170C2×C71#231011yescharacteristic
171C2×C71C22×#10211yescharacteristic
172C2×C71C2×#41911yescharacteristic
173C2×C71#322612yes—
174C2×C71#322912yes—
175C22×C4⋊Q8D7111yescharacteristic
176C2×#431—711no—
177#2285—711no—
178#1990—714no—
179#1988—714no—
180#1823—714no—
181C71#2183311yescharacteristic, complemented, Sylow
182C23×Q8—21no—
183C2×C4⋊Q8—21no—
184C2×C4⋊Q8—21no—
185C2×C4⋊Q8—24no—
186C2×C4⋊Q8Dic7118yescomplemented
187C2×C4⋊Q8C2×D7112yes—
188C2×C4⋊Q8C2×D7112yes—
189C2×#117—1421no—
190C22×C4⋊C4—21no—
191C22×C4⋊C4C2×D7112yes—
192C22×C42C2×D7111yescharacteristic
193#431—1424no—
194C2×C4⋊C8—712no—
195C2×C22⋊C8—712no—
196C2×C4×C8—712no—
197#384—712no—
198#382—1421no—
199#380—712no—
200#380—1421no—
201C4×C4⋊C4—712no—
202C4×C4⋊C4—1421no—
203#340—7116no—
204C22×Q8—28no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—44no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4C22⋊D7111yescharacteristic
211C23×C4C22⋊D7111yescharacteristic
212C23×C4C22×D7111yescharacteristic
213C22×C8—714no—
214C4⋊Q8—28no—
215C4⋊Q8—28no—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8C2×Dic7118yes—
219C4⋊Q8C2×Dic7118yes—
220#117—2844no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—714no—
227C2×C4⋊C4—1422no—
228C2×C4⋊C4—1422no—
229C2×C4⋊C4C2×Dic7118yes—
230C2×C4⋊C4C22×D7114yes—
231C2×C22⋊C4—1422no—
232C2×C42—1422no—
233C2×C42C2×Dic7114yes—
234C2×C42C22⋊D7111yescharacteristic
235C2×C42C22⋊D7111yescharacteristic
236C2×C42C22⋊D7114yes—
237C2×C42C22×D7111yescharacteristic
238C2×C42C22×D7111yescharacteristic
239C4⋊C8—1424no—
240C22⋊C8—1424no—
241C4×C8—1424no—
242#95—1422no—
243#95—1422no—
244C24C2×C22⋊D7111yescharacteristic
245C2×Q8—28no—
246C2×Q8—28no—
247C2×Q8—432no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—1424no—
259C22×C4C22⋊Dic7114yes—
260C22×C4C22⋊Dic7114yes—
261C22×C4#1176585014yes—
262C22×C4#1176585314yes—
263C22×C4C22×Dic7112yes—
264C22×C4C2×C22⋊D7112yes—
265C22×C4C2×C22⋊D7111yescharacteristic
266C22×C4C2×C22⋊D7111yescharacteristic
267C22×C4C2×C22⋊D7111yescharacteristic
268C22×C4C2×C22⋊D7111yescharacteristic
269C22×C4C2×C22⋊D7111yescharacteristic
270C22×C4C23×D7111yescharacteristic
271C2×C8—1424no—
272C2×C8—1428no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—2844no—
278C4⋊C4—2844no—
279C4⋊C4—1428no—
280C4⋊C4C22×Dic7118yes—
281C22⋊C4—2844no—
282C42—2844no—
283C42C22⋊Dic7114yes—
284C42C22⋊Dic7114yes—
285C42C22×Dic7112yes—
286C42C22×Dic7112yes—
287C42C2×C22⋊D7112yes—
288C42C2×C22⋊D7112yes—
289C23—22no—
290C23—22no—
291C23C2×C22⋊Dic7112yes—
292C23C2×#1176585012yes—
293C23C2×#1176585312yes—
294C23C22×C22⋊D7111yescharacteristic
295Q8—416no—
296Q8—416no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—2844no—
307C2×C4—2844no—
308C2×C4—2844no—
309C2×C4#14556049116yes—
310C2×C4C2×C22⋊Dic7112yes—
311C2×C4C2×C22⋊Dic7112yes—
312C2×C4C2×C22⋊Dic7112yes—
313C2×C4C2×C22⋊Dic7112yes—
314C2×C4C2×C22⋊Dic7112yes—
315C2×C4C2×#1176585012yes—
316C2×C4C2×#1176585012yes—
317C2×C4C2×#1176585312yes—
318C2×C4C2×#1176585312yes—
319C2×C4#1455615914yes—
320C2×C4C23×Dic7111yescharacteristic
321C2×C4C22×C22⋊D7111yescharacteristic
322C2×C4#1455617614yes—
323C8—2844no—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—22no—
328C22C2×#1455604914yes—
329C22C22×C22⋊Dic7111yescharacteristic
330C22#1798657312yes—
331C22C22×#1176585011yescharacteristic
332C22C22×#1176585311yescharacteristic
333C22C2×#1455615912yes—
334C4—22no—
335C4—22no—
336C4—48no—
337C4—5684no—
338C4C2×#1455604914yes—
339C4C2×#1455604914yes—
340C4#1798600814yes—
341C4#1798647712yes—
342C4#1798657312yes—
343C2—24no—
344C2#2259092611yescharacteristic
345C2C22×#1455604911yescharacteristic
346C2C2×#1798600811yescharacteristic
347C2#2259946112yes—
348C2#2259948412yes—
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×C4⋊Q8×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenterC22×C4⋊Q8D71
hypocenterC71#21833
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22589302C2semidirect
4. #22589293C2semidirect
23. C2×C4⋊Q8×C71C4semidirect
181. C71#21833semidirect
186. C2×C4⋊Q8Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860522
#225907472
#225893022
#225892932
#225884262
C22×C4⋊Q8×C712
#2183371