Finite Group Databasebeta

← #32053423

Subgroups — #32053423

Order 18176.

986 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986395C211yesmaximal, characteristic
2C2×#17986294C211yesmaximal, characteristic
3#22593545C211yesmaximal, characteristic
4#22589675C214yesmaximal
5#22589293C211yesmaximal, characteristic
6#22589277C211yesmaximal, characteristic
7#22588608C214yesmaximal
8C4×#14556052C211yesmaximal, characteristic
9C2×#557×C71C211yesmaximal, characteristic
10C2×#14556128C2211yescharacteristic
11#17986395—24no—
12#17986294—24no—
13C2×#14556061C2212yes—
14C2×#14556059C2214yes—
15C2×C4⋊(C71⋊C8)C2211yescharacteristic
16C2×#14556052C2212yes—
17C2×#14556052C2212yes—
18C2×C4×C71⋊C8C2211yescharacteristic
19#17985919—22no—
20#17985906—21no—
21#17985897C2214yes—
22#17985868—21no—
23C4×C4⋊Dic71C2211yescharacteristic
24#17985761C2212yes—
25#17985761C2214yes—
26#17985754C2214yes—
27#557×C71—24no—
28#557×C71—24no—
29C2×C4⋊Q8×C71—21no—
30C2×C4⋊Q8×C71C2211yescharacteristic
31C2×#125×C71—22no—
32C2×#125×C71C2212yes—
33C2×C4×Q8×C71—21no—
34C2×C4×Q8×C71C2212yes—
35C2×C4×Q8×C71C2211yescharacteristic
36C2×C4×Q8×C71C2211yescharacteristic
37C22×Dic142—21no—
38C22×C71⋊C8C2312yes—
39C22×C71⋊C8C2312yes—
40#14556128—22no—
41#14556128—22no—
42C2×C4⋊Dic71—21no—
43C2×C4⋊Dic71C2312yes—
44C2×C4⋊Dic71C2312yes—
45C2×Dic71⋊C4—24no—
46C2×C4×Dic71—21no—
47#14556061—24no—
48#14556059—28no—
49C4⋊(C71⋊C8)—22no—
50#14556052—24no—
51#14556052Q818yescomplemented
52C4×C71⋊C8—22no—
53C71⋊#95—21no—
54C71⋊#95—24no—
55C22×Q8×C71—21no—
56C22×Q8×C71D411yescharacteristic
57C22×Q8×C71C2311yescharacteristic
58C4⋊Q8×C71—28no—
59C4⋊Q8×C71—22no—
60C4⋊Q8×C71D414yes—
61#125×C71—216no—
62#125×C71—24no—
63#125×C71—24no—
64C4×Q8×C71—28no—
65C4×Q8×C71—24no—
66C4×Q8×C71—24no—
67C4×Q8×C71—22no—
68C4×Q8×C71—22no—
69C4×Q8×C71—22no—
70C4×Q8×C71D414yes—
71C2×C4⋊C4×C71—22no—
72C2×C4⋊C4×C71—21no—
73C2×C4⋊C4×C71—22no—
74C2×C4⋊C4×C71—21no—
75C2×C4⋊C4×C71—22no—
76C2×C4⋊C4×C71D411yescharacteristic
77C2×C4⋊C4×C71D412yes—
78C2×C4⋊C4×C71C2311yescharacteristic
79C2×C4⋊C4×C71C2312yes—
80C2×C42×C71—22no—
81C2×C42×C71—21no—
82C2×C42×C71C2312yes—
83C2×C42×C71C2311yescharacteristic
84C2×Dic142—42no—
85C2×Dic142—82no—
86C22×Dic71—22no—
87C22×Dic71—22no—
88C2×C71⋊C8—22no—
89C2×C71⋊C8—22no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8—24no—
92C2×C71⋊C8C2×Q814yes—
93C4⋊Dic71—22no—
94C4⋊Dic71—22no—
95C4⋊Dic71—22no—
96C4⋊Dic71—42no—
97C4⋊Dic71C2×Q814yes—
98Dic71⋊C4—84no—
99C4×Dic71—42no—
100C2×Q8×C71—22no—
101C2×Q8×C71—24no—
102C2×Q8×C71—22no—
103C2×Q8×C71—22no—
104C2×Q8×C71—21no—
105C2×Q8×C71—22no—
106C2×Q8×C71—42no—
107C2×Q8×C71—42no—
108C2×Q8×C71C2×D412yes—
109C2×Q8×C71C2×Q814yes—
110C22×C4×C71—21no—
111C22×C4×C71—21no—
112C22×C4×C71—22no—
113C22×C4×C71C2×D412yes—
114C22×C4×C71C2×D412yes—
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71C2411yescharacteristic
118C4⋊C4×C71—28no—
119C4⋊C4×C71—28no—
120C4⋊C4×C71—28no—
121C4⋊C4×C71—24no—
122C4⋊C4×C71—24no—
123C4⋊C4×C71—22no—
124C4⋊C4×C71—24no—
125C4⋊C4×C71—21no—
126C4⋊C4×C71—22no—
127C4⋊C4×C71C2×D414yes—
128C4⋊C4×C71C2×D412yes—
129C42×C71—28no—
130C42×C71—24no—
131C42×C71—22no—
132C42×C71—22no—
133C42×C71C2×D412yes—
134C42×C71C2×D412yes—
135C23×C71C22×D411yescharacteristic
136Q8×C71—22no—
137Q8×C71—24no—
138Q8×C71—44no—
139Q8×C71—48no—
140Q8×C71C4⋊Q814yes—
141C2×C4×C71—22no—
142C2×C4×C71—24no—
143C2×C4×C71—22no—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—22no—
147C2×C4×C71—24no—
148C2×C4×C71—22no—
149C2×C4×C71—22no—
150C2×C4×C71—24no—
151C2×C4×C71—42no—
152C2×C4×C71—42no—
153C2×C4×C71—42no—
154C2×C4×C71#12012yes—
155C2×C4×C71#12012yes—
156C2×C4×C71#13512yes—
157C2×C4×C71#13712yes—
158C2×C4×C71C22×D411yescharacteristic
159C2×C4×C71C22×D411yescharacteristic
160C2×C4×C71C22×Q812yes—
161C2×C4×C712−1+412yes—
162C2×Dic71—24no—
163C2×Dic71—42no—
164C2×Dic71—42no—
165C2×Dic71—42no—
166C2×Dic71—44no—
167Dic142—84no—
168C71⋊C8—24no—
169C71⋊C8—42no—
170C22×C71C2×#12011yescharacteristic
171C22×C71D4×Q812yes—
172C22×C71C2×#13511yescharacteristic
173C22×C71C2×#13711yescharacteristic
174C22×C71#57712yes—
175C4×C71—24no—
176C4×C71—22no—
177C4×C71—42no—
178C4×C71—44no—
179C4×C71—42no—
180C4×C71—42no—
181C4×C71#45214yes—
182C4×C71C2×C4⋊Q811yescharacteristic
183C4×C71#53611yescharacteristic
184C4×C71D4×Q812yes—
185Dic71—44no—
186Dic71—82no—
187#25836—711nomaximal, Sylow
188C2×C71#243411yescharacteristic
189C2×C71C2×#45211yescharacteristic
190C2×C71#301011yescharacteristic
191C2×C71#338112yes—
192C2×C71#338712yes—
193C2×#557D7111yescharacteristic
194C2×#498—711no—
195C2×#477—711no—
196#2567—711no—
197#2059—714no—
198#1988—711no—
199#1984—711no—
200#1865—714no—
201C4×#103—711no—
202C71#2583611yescharacteristic, complemented, Sylow
203#557—24no—
204#557—24no—
205C2×C4⋊Q8—21no—
206C2×C4⋊Q8—711no—
207C2×C4⋊Q8C2×D7111yescharacteristic
208C2×#125—22no—
209C2×#125C2×D7112yes—
210C2×C4×Q8—21no—
211C2×C4×Q8C2×D7112yes—
212C2×C4×Q8C2×D7111yescharacteristic
213C2×C4×Q8C2×D7111yescharacteristic
214#498—1424no—
215#477—1424no—
216C2×#107—712no—
217C2×#106—714no—
218C2×C4⋊C8—711no—
219C2×#103—712no—
220C2×#103—712no—
221C2×C4×C8—711no—
222#395—1422no—
223#391—1421no—
224#389—714no—
225#384—1421no—
226C4×C4⋊C4—711no—
227#340—712no—
228#340—714no—
229#336—714no—
230C22×Q8—21no—
231C22×Q8—1421no—
232C22×Q8C22⋊D7111yescharacteristic
233C22×Q8C22×D7111yescharacteristic
234C22×C8—712no—
235C22×C8—712no—
236C4⋊Q8—28no—
237C4⋊Q8—22no—
238C4⋊Q8—1422no—
239C4⋊Q8—1422no—
240C4⋊Q8C22⋊D7114yes—
241#125—216no—
242#125—24no—
243#125—24no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8C22⋊D7114yes—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—712no—
257C2×C4⋊C4—712no—
258C2×C4⋊C4—1421no—
259C2×C4⋊C4—1424no—
260C2×C4⋊C4C22⋊D7111yescharacteristic
261C2×C4⋊C4C22⋊D7112yes—
262C2×C4⋊C4C22×D7111yescharacteristic
263C2×C4⋊C4C22×D7112yes—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—1421no—
267C2×C42C22×D7112yes—
268C2×C42C22×D7111yescharacteristic
269#107—1424no—
270#106—1428no—
271C4⋊C8—1422no—
272#103—718no—
273#103—1424no—
274C4×C8—1422no—
275#95—1421no—
276#95—1424no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—2842no—
286C2×Q8—5682no—
287C2×Q8C2×C22⋊D7112yes—
288C2×Q8Q8×D7114yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—1422no—
293C22×C4—1422no—
294C22×C4C2×C22⋊D7112yes—
295C22×C4C2×C22⋊D7112yes—
296C22×C4C2×C22⋊D7111yescharacteristic
297C22×C4C2×C22⋊D7111yescharacteristic
298C22×C4C23×D7111yescharacteristic
299C2×C8—714no—
300C2×C8—1422no—
301C2×C8—1424no—
302C2×C8—1422no—
303C2×C8—1422no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—2842no—
314C4⋊C4—5684no—
315C4⋊C4—714no—
316C4⋊C4—1422no—
317C4⋊C4—1422no—
318C4⋊C4—1422no—
319C4⋊C4C2×C22⋊D7114yes—
320C4⋊C4C2×C22⋊D7112yes—
321C42—28no—
322C42—22no—
323C42—22no—
324C42—24no—
325C42—2842no—
326C42C2×C22⋊D7112yes—
327C42C2×C22⋊D7112yes—
328C23C22×C22⋊D7111yescharacteristic
329Q8—22no—
330Q8—24no—
331Q8—44no—
332Q8—48no—
333Q8—5684no—
334Q8#1455612614yes—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—2842no—
349C2×C4—2842no—
350C2×C4—2842no—
351C2×C4—2844no—
352C2×C4—1424no—
353C2×C4#1455609712yes—
354C2×C4#1455609712yes—
355C2×C4#1455614812yes—
356C2×C4#1455615812yes—
357C2×C4C22×C22⋊D7111yescharacteristic
358C2×C4C22×C22⋊D7111yescharacteristic
359C2×C4C2×Q8×D7112yes—
360C2×C4#1455617612yes—
361C8—2842no—
362C8—1424no—
363C22C2×#1455609711yescharacteristic
364C22Q8×C22⋊D7112yes—
365C22C2×#1455614811yescharacteristic
366C22C2×#1455615811yescharacteristic
367C22#1798680712yes—
368C4—24no—
369C4—22no—
370C4—42no—
371C4—44no—
372C4—42no—
373C4—42no—
374C4—2844no—
375C4—5682no—
376C4#1798616114yes—
377C4C2×#1455612611yescharacteristic
378C4#1798657311yescharacteristic
379C4Q8×C22⋊D7112yes—
380C2#2259198011yescharacteristic
381C2C2×#1798616111yescharacteristic
382C2#2259762611yescharacteristic
383C2#2260121112yes—
384C2#2260127212yes—
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#557×C71C2
socle, prime radicalC23×C71C22×D4
hypercenterC2×#557D71
hypocenterC71#25836
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
51. #14556052Q8semidirect
202. C71#25836semidirect

Maximal subgroups

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

subgroupindex
C2×#179863952
C2×#179862942
#225935452
#225896752
#225892932
#225892772
#225886082
C4×#145560522
C2×#557×C712
#2583671