Finite Group Databasebeta

← #31891800

Subgroups — #31891800

Order 18176.

376 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1M16×Dic142C211yesmaximal, characteristic
2#22587095C211yesmaximal, characteristic
3#22586768C211yesmaximal, characteristic
4C2×#17985729C211yesmaximal, characteristic
5#22586237C211yesmaximal, characteristic
6#22586178C211yesmaximal, characteristic
7C71×#1950C211yesmaximal, characteristic
8C2×C4×Dic142C2211yescharacteristic
9#17986129—21no—
10#17986127—22no—
11C8×Dic142—21no—
12#17986052C2211yescharacteristic
13#17986023C2211yescharacteristic
14#17986022—21no—
15M16×Dic71—21no—
16C2×C4×C71⋊C8C2211yescharacteristic
17#17985749—22no—
18#17985729—21no—
19#17985729—21no—
20#17985729C411yescharacteristic
21#17985729C411yescharacteristic
22#17985729C411yescharacteristic
23#17985729C411yescharacteristic
24C2×#108×C71—21no—
25#423×C71C2211yescharacteristic
26C2×C4⋊C8×C71C2211yescharacteristic
27C4×M16×C71C2211yescharacteristic
28#341×C71—21no—
29C22×Dic142Q811yescharacteristic
30C2×#11765849—21no—
31C22×C71⋊C8—21no—
32C4×Dic142—21no—
33C4×Dic142—21no—
34C4×Dic142C2×C411yescharacteristic
35C4×Dic142C2×C411yescharacteristic
36C4×Dic142D411yescharacteristic
37C4×Dic142D411yescharacteristic
38C2×C4⋊Dic71Q811yescharacteristic
39C2×Dic71⋊C4—21no—
40C2×C4×Dic71—21no—
41C4⋊(C71⋊C8)—21no—
42C4⋊(C71⋊C8)—21no—
43C4⋊(C71⋊C8)D412yes—
44Dic71⋊C8—22no—
45Dic71⋊C8—22no—
46C22⋊(C71⋊C8)—21no—
47#14556038—22no—
48C4×C71⋊C8—21no—
49C4×C71⋊C8—21no—
50C4×C71⋊C8C2×C411yescharacteristic
51C4×C71⋊C8C2×C411yescharacteristic
52C8×Dic71—22no—
53C2×M16×C71—21no—
54C2×M16×C71D411yescharacteristic
55C2×M16×C71D411yescharacteristic
56C22×C8×C71—21no—
57C2×C42×C71C2311yescharacteristic
58#108×C71—42no—
59C4⋊C8×C71—21no—
60C4⋊C8×C71—21no—
61C4⋊C8×C71—21no—
62C4⋊C8×C71C2×C411yescharacteristic
63C4⋊C8×C71C2×C411yescharacteristic
64#97×C71—21no—
65C4×C8×C71—21no—
66C2×Dic142—22no—
67C2×Dic142—21no—
68C2×Dic142—41no—
69C2×Dic142C4⋊C411yescharacteristic
70C2×Dic142C4⋊C411yescharacteristic
71C22×Dic71—21no—
72C22×Dic71—21no—
73#11765849—41no—
74#11765849—41no—
75C2×C71⋊C8—21no—
76C2×C71⋊C8—21no—
77C2×C71⋊C8—22no—
78C2×C71⋊C8—42no—
79C2×C71⋊C8—41no—
80C4⋊Dic71—21no—
81C4⋊Dic71C4⋊C411yescharacteristic
82C4⋊Dic71C4⋊C411yescharacteristic
83Dic71⋊C4—21no—
84Dic71⋊C4—21no—
85Dic71⋊C4—21no—
86Dic71⋊C4—21no—
87C4×Dic71—21no—
88C4×Dic71—21no—
89C4×Dic71—21no—
90C4×Dic71—21no—
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×Q811yescharacteristic
93C22×C4×C71#4111yescharacteristic
94M16×C71—21no—
95M16×C71—21no—
96M16×C71—21no—
97M16×C71—41no—
98M16×C71—41no—
99M16×C71SD1612yes—
100C2×C8×C71—21no—
101C2×C8×C71—21no—
102C2×C8×C71—21no—
103C2×C8×C71—21no—
104C2×C8×C71—41no—
105C2×C8×C71—41no—
106C2×C8×C71—41no—
107C42×C71C22×C411yescharacteristic
108C42×C71C2×D411yescharacteristic
109C42×C71C2×D411yescharacteristic
110C42×C71#4111yescharacteristic
111C23×C71C22⋊Q811yescharacteristic
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—41no—
115C2×C4×C71C2×C4⋊C411yescharacteristic
116C2×C4×C71C4×D411yescharacteristic
117C2×C4×C71C4×D411yescharacteristic
118C2×C4×C71#12011yescharacteristic
119C2×C4×C71C22⋊Q811yescharacteristic
120C2×C4×C71#12311yescharacteristic
121C2×C4×C71C2×SD1611yescharacteristic
122C2×C4×C71#13511yescharacteristic
123C2×C4×C71#13711yescharacteristic
124C2×C4×C71#13711yescharacteristic
125C2×Dic71—21no—
126C2×Dic71—21no—
127C2×Dic71—21no—
128C2×Dic71—21no—
129C2×Dic71—42no—
130C2×Dic71—81no—
131Dic142—21no—
132Dic142—21no—
133Dic142—21no—
134Dic142—42no—
135Dic142#10812yescomplemented
136C8×C71—22no—
137C8×C71—41no—
138C8×C71—41no—
139C8×C71—41no—
140C8×C71—81no—
141C71⋊C8—22no—
142C71⋊C8—41no—
143C71⋊C8—42no—
144C71⋊C8—81no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71#38511yescharacteristic
148C22×C71#47511yescharacteristic
149C22×C71#47911yescharacteristic
150C4×C71—21no—
151C4×C71—21no—
152C4×C71—21no—
153C4×C71C2×#10811yescharacteristic
154C4×C71#43011yescharacteristic
155C4×C71C4×SD1611yescharacteristic
156C4×C71#44111yescharacteristic
157C4×C71#44811yescharacteristic
158C4×C71#45211yescharacteristic
159Dic71—22no—
160Dic71—41no—
161Dic71—41no—
162Dic71—41no—
163#8909—711nomaximal, Sylow
164C2×C71—21no—
165C2×C71—21no—
166C2×C71#185511yescharacteristic
167C2×C71#186111yescharacteristic
168C2×C71#186511yescharacteristic
169Q8×M16—711no—
170#1950D7111yescharacteristic
171#1566—711no—
172#1487—711no—
173C2×Q8⋊C8—711no—
174#1301—711no—
175#1278—711no—
176C71#890911yescharacteristic, complemented, Sylow
177C2×C4×Q8—711no—
178#446—1421no—
179#446—1422no—
180C8×Q8—1421no—
181#431—711no—
182C2×#108—21no—
183#423—711no—
184#423—1421no—
185#423C2×D7111yescharacteristic
186C2×C4⋊C8C2×D7111yescharacteristic
187C4×M16—1421no—
188C4×M16C2×D7111yescharacteristic
189C2×C4×C8—711no—
190#341—21no—
191#334—1422no—
192Q8⋊C8—711no—
193Q8⋊C8—711no—
194Q8⋊C8—711no—
195Q8⋊C8—711no—
196Q8⋊C8—1421no—
197Q8⋊C8—1421no—
198C22×Q8—711no—
199C2×M16—21no—
200C2×M16—1421no—
201C2×M16C22⋊D7111yescharacteristic
202C2×M16C22⋊D7111yescharacteristic
203C22×C8—21no—
204C22×C8—1421no—
205C4×Q8—711no—
206C4×Q8—711no—
207C4×Q8—711no—
208C4×Q8—711no—
209C4×Q8—1421no—
210C4×Q8—1421no—
211C2×C4⋊C4—711no—
212C2×C4⋊C4—1421no—
213C2×C42—1421no—
214C2×C42C22×D7111yescharacteristic
215#108—42no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—712no—
220C4⋊C8—1422no—
221C4⋊C8—1422no—
222C4⋊C8—1421no—
223C4⋊C8—1421no—
224C4⋊C8C4×D7111yescharacteristic
225C4⋊C8C4×D7111yescharacteristic
226C22⋊C8—1421no—
227#97—21no—
228#97—1422no—
229C4×C8—21no—
230C4×C8—711no—
231C4×C8—711no—
232C4×C8—1422no—
233C4×C8—1421no—
234C4×C8—1421no—
235C2×Q8—2841no—
236C2×Q8—711no—
237C2×Q8—711no—
238C2×Q8—1422no—
239C2×Q8—1421no—
240C22×C4—1421no—
241C22×C4—1421no—
242C22×C4C2×C22⋊D7111yescharacteristic
243C22×C4Q8×D7111yescharacteristic
244C22×C4#1176586611yescharacteristic
245M16—21no—
246M16—21no—
247M16—21no—
248M16—41no—
249M16—41no—
250M16—2841no—
251M16—2841no—
252M16#1176585312yes—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—2842no—
261C2×C8—2841no—
262C2×C8—1421no—
263C2×C8—1422no—
264C2×C8—1421no—
265C4⋊C4—711no—
266C4⋊C4—711no—
267C4⋊C4—1421no—
268C4⋊C4—1421no—
269C4⋊C4—1421no—
270C4⋊C4—1421no—
271C4⋊C4—1421no—
272C42—1421no—
273C42—1421no—
274C42—1421no—
275C42—1421no—
276C42C2×C4×D7111yescharacteristic
277C42D4×D7111yescharacteristic
278C42D4×D7111yescharacteristic
279C42#1176586511yescharacteristic
280C23#1455610411yescharacteristic
281Q8—2842no—
282Q8—712no—
283Q8—1421no—
284Q8—1421no—
285Q8—1421no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—2842no—
290C2×C4—5681no—
291C2×C4—1421no—
292C2×C4—1421no—
293C2×C4—1421no—
294C2×C4—1421no—
295C2×C4C4⋊C4×D7111yescharacteristic
296C2×C4C4×C22⋊D7111yescharacteristic
297C2×C4#1455609011yescharacteristic
298C2×C4#1455609611yescharacteristic
299C2×C4#1455610611yescharacteristic
300C2×C4#1455611311yescharacteristic
301C2×C4C2×#1176585311yescharacteristic
302C2×C4#1455614811yescharacteristic
303C2×C4#1455615811yescharacteristic
304C2×C4#1455615911yescharacteristic
305C8—22no—
306C8—41no—
307C8—41no—
308C8—41no—
309C8—81no—
310C8—2841no—
311C8—2842no—
312C8—5681no—
313C8—1422no—
314C22—21no—
315C22—21no—
316C22#1798587011yescharacteristic
317C22#1798628111yescharacteristic
318C22#1798630311yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—2841no—
323C4—2841no—
324C4—2841no—
325C4—1422no—
326C4#108×D7111yescharacteristic
327C4#1798604611yescharacteristic
328C4C4×#1176585311yescharacteristic
329C4#1798610411yescharacteristic
330C4#1798613711yescharacteristic
331C4#1798615911yescharacteristic
332C2—21no—
333C2—21no—
334C2#2258854311yescharacteristic
335C2#2258857711yescharacteristic
336C2#2258860711yescharacteristic
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1950C2
socleC22×C71#385
prime radicalC23×C71C22⋊Q8
cosocleC2×C42×C71C23
hypercenter#1950D71
hypocenterC71#8909
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
135. Dic142#108semidirect
176. C71#8909semidirect

Maximal subgroups

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

subgroupindex
M16×Dic1422
#225870952
#225867682
C2×#179857292
#225862372
#225861782
C71×#19502
#890971