Finite Group Databasebeta

← #31916604

Subgroups — #31916604

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985824C211yesmaximal, characteristic, complemented
2C2×#17985800C211yesmaximal, characteristic, complemented
3C2×#17985800C211yesmaximal, characteristic, complemented
4C2×C22⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
5#22588369C211yesmaximal, characteristic, complemented
6#22586431C211yesmaximal, characteristic, complemented
7C2×#466×C71C211yesmaximal, characteristic
8C22×C71⋊C16C2211yescharacteristic
9C22×C71⋊C16C2211yescharacteristic
10C2×#14556062C2211yescharacteristic
11C2×#14556062C2211yescharacteristic
12#17985859—21no—
13#17985824—22no—
14#17985800—22no—
15#17985800—22no—
16C22⋊(C71⋊C16)—22no—
17#17985762—21no—
18C22×D8×C71C2211yescharacteristic
19C23×C8×C71C2211yescharacteristic
20C2×#121×C71—21no—
21#466×C71—22no—
22#466×C71—22no—
23C2×#107×C71C2211yescharacteristic
24C2×#102×C71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×C22⋊Dic71—41no—
28C2×C71⋊C16—21no—
29C2×C71⋊C16—21no—
30C2×C71⋊C16—21no—
31C2×C71⋊C16—21no—
32C2×C71⋊C16—21no—
33C2×C71⋊C16D412yescomplemented
34#14556062—21no—
35#14556062—21no—
36#14556062—21no—
37#14556062—21no—
38C71⋊#95—41no—
39C22×D4×C71—21no—
40C22×D4×C71—41no—
41C23×C4×C71D411yescharacteristic
42C2×D8×C71—21no—
43C2×D8×C71—21no—
44C2×D8×C71—21no—
45C2×D8×C71—21no—
46C2×D8×C71—21no—
47C2×D8×C71D412yes—
48C22×C8×C71—21no—
49C22×C8×C71—21no—
50C22×C8×C71—21no—
51C22×C8×C71—21no—
52C22×C8×C71—21no—
53C22×C8×C71—21no—
54C22×C8×C71D411yescharacteristic
55C22×C8×C71C2311yescharacteristic
56#121×C71—24no—
57#121×C71—42no—
58C2×C4⋊C4×C71—21no—
59C2×C22⋊C4×C71—41no—
60#107×C71—21no—
61#107×C71D412yes—
62#102×C71—24no—
63C22×Dic71—41no—
64C22×Dic71—41no—
65C4⋊Dic71—41no—
66C4⋊Dic71—41no—
67C4⋊Dic71—41no—
68C4⋊Dic71—41no—
69C22⋊Dic71—82no—
70C71⋊C16—22no—
71C71⋊C16—41no—
72C24×C71—41no—
73C24×C71D811yescharacteristic
74C2×D4×C71—22no—
75C2×D4×C71—44no—
76C2×D4×C71—41no—
77C2×D4×C71—41no—
78C2×D4×C71—41no—
79C2×D4×C71—41no—
80C2×D4×C71—41no—
81C2×D4×C71—82no—
82C2×D4×C71—82no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—41no—
90C22×C4×C71D811yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92D8×C71—22no—
93D8×C71—22no—
94D8×C71—41no—
95D8×C71—41no—
96C2×C8×C71—21no—
97C2×C8×C71—21no—
98C2×C8×C71—21no—
99C2×C8×C71—41no—
100C2×C8×C71—41no—
101C2×C8×C71—41no—
102C2×C8×C71—41no—
103C2×C8×C71C2×D411yescharacteristic
104C2×C8×C71C2×D411yescharacteristic
105C2×C8×C71C2×D411yescharacteristic
106C2×C8×C71#4111yescharacteristic
107C2×C8×C71#4111yescharacteristic
108C2×C8×C71#4111yescharacteristic
109C4⋊C4×C71—22no—
110C4⋊C4×C71—41no—
111C22⋊C4×C71—44no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71—42no—
119C23×C71—81no—
120C23×C71—81no—
121C23×C71—81no—
122C23×C71—81no—
123C23×C71—81no—
124C23×C71—81no—
125C23×C71D1612yes—
126C23×C71C2×D811yescharacteristic
127D4×C71—42no—
128D4×C71—42no—
129D4×C71—88no—
130D4×C71—81no—
131D4×C71—81no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—42no—
140C2×C4×C71—81no—
141C2×C4×C71—81no—
142C2×C4×C71#12011yescharacteristic
143C2×C4×C71#12111yescharacteristic
144C2×C4×C71#12311yescharacteristic
145C2×C4×C71#13511yescharacteristic
146C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C71#13711yescharacteristic
148C2×Dic71—81no—
149C2×Dic71—81no—
150C2×Dic71—81no—
151C2×Dic71—81no—
152C2×Dic71—81no—
153C2×Dic71—81no—
154C8×C71—41no—
155C8×C71#12111yescharacteristic
156C8×C71#12111yescharacteristic
157C8×C71#12311yescharacteristic
158C8×C71#12411yescharacteristic
159C22×C71—22no—
160C22×C71—22no—
161C22×C71—22no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—82no—
167C22×C71—82no—
168C22×C71—82no—
169C22×C71—161no—
170C22×C71—161no—
171C22×C71—161no—
172C22×C71—161no—
173C22×C71#44711yescharacteristic
174C22×C71#46611yescharacteristic
175C22×C71#48011yescharacteristic
176C22×C71C2×D1611yescharacteristic
177C22×C71#50811yescharacteristic
178C22×C71#50911yescharacteristic
179C22×C71#51011yescharacteristic
180C4×C71—41no—
181C4×C71—82no—
182C4×C71#39411yescharacteristic
183C4×C71#45211yescharacteristic
184C4×C71#46811yescharacteristic
185C4×C71#48211yescharacteristic
186Dic71—161no—
187Dic71—161no—
188#13380—711nomaximal, Sylow
189C2×C71—42no—
190C2×C71—162no—
191C2×C71#203311yescharacteristic
192C2×C71#218411yescharacteristic
193C2×C71#218911yescharacteristic
194C2×C71#221511yescharacteristic
195C2×C71#222011yescharacteristic
196C2×C71#223211yescharacteristic
197C2×C71#223411yescharacteristic
198C2×#466D7111yescharacteristic
199C2×#366—711no—
200C2×#357—711no—
201C2×#357—711no—
202C2×C22⋊C16—711no—
203#1808—711no—
204#1380—711no—
205C71#1338011yescharacteristic, complemented, Sylow
206C22×D8C2×D7111yescharacteristic
207C23×C8C2×D7111yescharacteristic
208C2×#121—21no—
209C22×C16—711no—
210C22×C16—711no—
211#466—22no—
212#466—22no—
213C2×#107—711no—
214C2×#107—711no—
215C2×#107C2×D7111yescharacteristic
216C2×#102—21no—
217#380—1421no—
218#366—1422no—
219#357—1422no—
220#357—1422no—
221C22⋊C16—1422no—
222#340—1421no—
223C22×D4—21no—
224C22×D4—41no—
225C23×C4C22⋊D7111yescharacteristic
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8C22⋊D7112yes—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8C22⋊D7111yescharacteristic
239C22×C8C22×D7111yescharacteristic
240#121—24no—
241#121—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1421no—
244C2×C4⋊C4—1421no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—2841no—
247C2×C16—712no—
248C2×C16—1421no—
249C2×C16—1421no—
250C2×C16—1421no—
251C2×C16—1421no—
252C2×C16—1421no—
253#107—21no—
254#107—1421no—
255#107—1421no—
256#107—1421no—
257#107—1421no—
258#107C22⋊D7112yes—
259#102—24no—
260#95—2841no—
261C24—41no—
262C24#1176585011yescharacteristic
263C2×D4—22no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—2841no—
280C22×C4—2841no—
281C22×C4#1176585011yescharacteristic
282C22×C4C2×C22⋊D7111yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D7111yescharacteristic
295C2×C8C2×C22⋊D7111yescharacteristic
296C2×C8C2×C22⋊D7111yescharacteristic
297C2×C8#1176586411yescharacteristic
298C2×C8#1176586411yescharacteristic
299C2×C8#1176586411yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—2841no—
303C4⋊C4—2841no—
304C4⋊C4—2841no—
305C4⋊C4—2841no—
306C22⋊C4—44no—
307C22⋊C4—5682no—
308C16—2841no—
309C16—1422no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23#1455606912yes—
324C23C2×#1176585011yescharacteristic
325D4—42no—
326D4—42no—
327D4—88no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—5681no—
341C2×C4—5681no—
342C2×C4—5681no—
343C2×C4—5681no—
344C2×C4—5681no—
345C2×C4—5681no—
346C2×C4#1455609711yescharacteristic
347C2×C4#1455610011yescharacteristic
348C2×C4#1455610911yescharacteristic
349C2×C4#1455614811yescharacteristic
350C2×C4#1455615411yescharacteristic
351C2×C4#1455615811yescharacteristic
352C8—41no—
353C8#1455609911yescharacteristic
354C8#1455610011yescharacteristic
355C8#1455610911yescharacteristic
356C8#1455611511yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22#1798613311yescharacteristic
372C22#1798623311yescharacteristic
373C22#1798631011yescharacteristic
374C22C2×#1455606911yescharacteristic
375C22#1798642911yescharacteristic
376C22#1798643511yescharacteristic
377C22#1798643911yescharacteristic
378C4—41no—
379C4—82no—
380C4—11361no—
381C4—11361no—
382C4#1798591311yescharacteristic
383C4#1798616111yescharacteristic
384C4#1798624311yescharacteristic
385C4#1798632011yescharacteristic
386C2—42no—
387C2—162no—
388C2#2258951911yescharacteristic
389C2#2259025211yescharacteristic
390C2#2259028011yescharacteristic
391C2#2259042411yescharacteristic
392C2#2259045111yescharacteristic
393C2#2259051811yescharacteristic
394C2#2259052811yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
Fitting, prime radicalC2×#466×C71C2
socleC23×C71C2×D8
hypercenterC2×#466D71
hypocenterC71#13380
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#17985824C2semidirect
2. C2×#17985800C2semidirect
3. C2×#17985800C2semidirect
4. C2×C22⋊(C71⋊C16)C2semidirect
5. #22588369C2semidirect
6. #22586431C2semidirect
33. C2×C71⋊C16D4semidirect
205. C71#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#179858242
C2×#179858002
C2×C22⋊(C71⋊C16)2
#225883692
#225864312
C2×#466×C712
#1338071