Finite Group Databasebeta

← #31892859

Subgroups — #31892859

Order 18176.

604 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986056C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic
3C2×#17985725C212yesmaximal
4#22586183C212yesmaximal
5C2×#387×C71C211yesmaximal, characteristic
6#17986056—24no—
7C2×C4⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)C2212yes—
9C2×C22⋊(C71⋊C8)C2212yes—
10C2×#14556039—21no—
11#17985754—21no—
12#17985754—21no—
13#17985725—24no—
14C22×C4⋊C4×C71C411yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16C22×C42×C71C2211yescharacteristic
17#387×C71—21no—
18#387×C71—21no—
19#387×C71—21no—
20#387×C71—21no—
21C2×#95×C71C411yescharacteristic
22C2×#95×C71C411yescharacteristic
23C2×#95×C71C411yescharacteristic
24C2×#95×C71C2211yescharacteristic
25C22×C71⋊C8—22no—
26C22×C71⋊C8—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)D414yes—
32#14556039—24no—
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—22no—
48C2×C42×C71—22no—
49C2×C42×C71—22no—
50C2×C42×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52C2×C42×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62C2×C71⋊C8—24no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8—42no—
68C24×C71C22×C411yescharacteristic
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—22no—
79C22×C4×C71—22no—
80C22×C4×C71—22no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71—41no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71—41no—
93C22×C4×C71—41no—
94C22×C4×C71C22⋊C411yescharacteristic
95C22×C4×C71C22⋊C411yescharacteristic
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71M1612yes—
99C22×C4×C71M1612yes—
100C22×C4×C71C2×D412yes—
101C22×C4×C71C2×D411yescharacteristic
102C22×C4×C71C2×D411yescharacteristic
103C22×C4×C71#4112yes—
104C4⋊C4×C71—24no—
105C4⋊C4×C71—24no—
106C4⋊C4×C71—41no—
107C4⋊C4×C71—42no—
108C4⋊C4×C71—41no—
109C42×C71—21no—
110C42×C71—21no—
111C42×C71—22no—
112C42×C71—22no—
113C42×C71—21no—
114C42×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71—21no—
119C23×C71C2×C22⋊C411yescharacteristic
120C23×C71C4×D412yes—
121C23×C71C2×M1611yescharacteristic
122C23×C71C2×M1611yescharacteristic
123C23×C71#13112yes—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—22no—
136C2×C4×C71—22no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—24no—
142C2×C4×C71—42no—
143C2×C4×C71—42no—
144C2×C4×C71—44no—
145C2×C4×C71—42no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—42no—
151C2×C4×C71—42no—
152C2×C4×C71—42no—
153C2×C4×C71—42no—
154C2×C4×C71—41no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—41no—
160C2×C4×C71—82no—
161C2×C4×C71—82no—
162C2×C4×C71#10112yes—
163C2×C4×C71#12011yescharacteristic
164C2×C4×C71#12112yes—
165C2×C4×C71#12311yescharacteristic
166C71⋊C8—44no—
167C71⋊C8—82no—
168C22×C71—22no—
169C22×C71—22no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—22no—
175C22×C71—22no—
176C22×C71#38611yescharacteristic
177C22×C71#40711yescharacteristic
178C22×C71#40811yescharacteristic
179C22×C71#41011yescharacteristic
180C22×C71#41011yescharacteristic
181C22×C71C2×#10111yescharacteristic
182C22×C71#41311yescharacteristic
183C22×C71#43512yes—
184C22×C71#43612yes—
185C4×C71—21no—
186C4×C71—22no—
187C4×C71—21no—
188C4×C71—22no—
189C4×C71—41no—
190C4×C71—41no—
191C4×C71—42no—
192C4×C71—41no—
193C4×C71—42no—
194C4×C71—81no—
195C4×C71—81no—
196#9086—711nomaximal, Sylow
197C2×C71—22no—
198C2×C71—22no—
199C2×C71#146411yescharacteristic
200C2×C71#146711yescharacteristic
201C2×C71#147011yescharacteristic
202C2×C71#147311yescharacteristic
203C2×C71#187011yescharacteristic
204C2×C71#189911yescharacteristic
205C2×C71#190111yescharacteristic
206C2×#432—711no—
207C2×#387D7111yescharacteristic
208#1817—711no—
209C2×#324—712no—
210#1280—712no—
211C71#908611yescharacteristic, complemented, Sylow
212C22×C4⋊C4Dic7111yescharacteristic
213C22×C4⋊C4C2×D7111yescharacteristic
214C22×C42C2×D7111yescharacteristic
215#432—1424no—
216C2×C4⋊C8—1421no—
217C2×C22⋊C8—712no—
218C2×C22⋊C8—712no—
219C2×#97—1421no—
220#387—21no—
221#387—21no—
222#387—21no—
223#387—21no—
224C2×#95Dic7111yescharacteristic
225C2×#95Dic7111yescharacteristic
226C2×#95Dic7111yescharacteristic
227C2×#95C2×D7111yescharacteristic
228#336—1421no—
229#336—1421no—
230#324—1424no—
231C23×C4C2×Dic7111yescharacteristic
232C23×C4C2×Dic7111yescharacteristic
233C23×C4C2×Dic7111yescharacteristic
234C23×C4C2×Dic7111yescharacteristic
235C23×C4C2×Dic7111yescharacteristic
236C23×C4C2×Dic7111yescharacteristic
237C23×C4C22×D7111yescharacteristic
238C22×C8—1422no—
239C22×C8—1422no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C22⋊D7111yescharacteristic
251C2×C42C22⋊D7111yescharacteristic
252C2×C42C22⋊D7111yescharacteristic
253C2×C42C22⋊D7111yescharacteristic
254C4⋊C8—1424no—
255C22⋊C8—714no—
256C22⋊C8—1422no—
257C22⋊C8—1422no—
258C22⋊C8—1422no—
259#97—1424no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268C24C22×Dic7111yescharacteristic
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C22⋊Dic7111yescharacteristic
295C22×C4C22⋊Dic7111yescharacteristic
296C22×C4C22⋊Dic7111yescharacteristic
297C22×C4C22⋊Dic7111yescharacteristic
298C22×C4#1176584912yes—
299C22×C4#1176584912yes—
300C22×C4D4×D7112yes—
301C22×C4C2×C22⋊D7111yescharacteristic
302C22×C4C2×C22⋊D7111yescharacteristic
303C22×C4#1176586412yes—
304C2×C8—2842no—
305C2×C8—2842no—
306C2×C8—2842no—
307C2×C8—2842no—
308C2×C8—2842no—
309C2×C8—1424no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23C2×C22⋊Dic7111yescharacteristic
326C23D4×Dic7112yes—
327C23C2×#1176584911yescharacteristic
328C23C2×#1176584911yescharacteristic
329C23#1455613512yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#1455604712yes—
369C2×C4#1455609611yescharacteristic
370C2×C4#1455610012yes—
371C2×C4#1455610911yescharacteristic
372C8—2844no—
373C8—5682no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22#1798587911yescharacteristic
383C22#1798596311yescharacteristic
384C22#1798596611yescharacteristic
385C22#1798597311yescharacteristic
386C22#1798597311yescharacteristic
387C22C2×#1455604711yescharacteristic
388C22#1798598311yescharacteristic
389C22#1798607112yes—
390C22#1798607712yes—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—22no—
395C4—42no—
396C4—41no—
397C4—42no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—81no—
402C2—22no—
403C2—22no—
404C2#2258667411yescharacteristic
405C2#2258668411yescharacteristic
406C2#2258669611yescharacteristic
407C2#2258670911yescharacteristic
408C2#2258863811yescharacteristic
409C2#2258879911yescharacteristic
410C2#2258880911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutator, prime radicalC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#387×C71C2
socleC23×C71C2×C22⋊C4
cosocleC23×C4×C71C23
hypercenterC2×#387D71
hypocenterC71#9086
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#387

Complements

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

NH ≅ G/Ntype
211. C71#9086semidirect

Maximal subgroups

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

subgroupindex
C2×#179860562
#225884082
C2×#179857252
#225861832
C2×#387×C712
#908671