Finite Group Databasebeta

← #31907659

Subgroups — #31907659

Order 18176.

514 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985768C211yesmaximal, characteristic, complemented
2C2×#17985725C211yesmaximal, characteristic, complemented
3#22586346C212yesmaximal, complemented
4#22586343C212yesmaximal, complemented
5C71×#2618C211yesmaximal, characteristic
6C22×#11765849—21no—
7C2×C22⋊(C71⋊C8)—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725—21no—
11#17985725C2212yes—
12C2×#123×C71—21no—
13C2×#121×C71C411yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#398×C71C411yescharacteristic
16#398×C71C2211yescharacteristic
17#396×C71—21no—
18#396×C71—21no—
19#394×C71C411yescharacteristic
20#394×C71C2211yescharacteristic
21#386×C71—21no—
22#380×C71C411yescharacteristic
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C22×C71⋊C8—41no—
28C22×C71⋊C8—41no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—41no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C411yescharacteristic
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C23×C4×C71D411yescharacteristic
36#123×C71—42no—
37#123×C71—42no—
38#121×C71—41no—
39#121×C71—41no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—41no—
44C2×C4⋊C4×C71—41no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71C2311yescharacteristic
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71C2×C411yescharacteristic
59#95×C71C2×C411yescharacteristic
60#11765849—24no—
61#11765849—44no—
62#11765849—42no—
63C2×C71⋊C8—44no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—82no—
67C24×C71—21no—
68C24×C71C22⋊C411yescharacteristic
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—42no—
73C2×D4×C71—42no—
74C2×D4×C71—81no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—22no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—81no—
86C22×C4×C71—81no—
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71D812yes—
91C22×C4×C71SD1612yes—
92C22×C4×C71C22×C411yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C22×C4×C71C2×D411yescharacteristic
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—42no—
101C4⋊C4×C71—41no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—81no—
104C22⋊C4×C71—42no—
105C22⋊C4×C71—42no—
106C22⋊C4×C71—42no—
107C22⋊C4×C71—41no—
108C22⋊C4×C71—41no—
109C22⋊C4×C71—81no—
110C22⋊C4×C71—81no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71—41no—
115C23×C71—41no—
116C23×C71—41no—
117C23×C71—41no—
118C23×C71—41no—
119C23×C71—41no—
120C23×C71#9911yescharacteristic
121C23×C71#9911yescharacteristic
122C23×C71#10212yes—
123C23×C71#10412yes—
124C23×C71C2×C22⋊C411yescharacteristic
125D4×C71—82no—
126D4×C71—82no—
127D4×C71—82no—
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—42no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—41no—
154C2×C4×C71—81no—
155C2×C4×C71—81no—
156C2×C4×C71—81no—
157C2×C4×C71#10212yes—
158C2×C4×C71C2×D811yescharacteristic
159C2×C4×C71C2×SD1611yescharacteristic
160C71⋊C8—44no—
161C71⋊C8—82no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—42no—
169C22×C71—42no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71—81no—
173C22×C71—81no—
174C22×C71—81no—
175C22×C71—81no—
176C22×C71—81no—
177C22×C71#32712yes—
178C22×C71#32712yes—
179C22×C71C2×#9911yescharacteristic
180C22×C71C2×#10211yescharacteristic
181C22×C71C2×#10411yescharacteristic
182C4×C71—21no—
183C4×C71—22no—
184C4×C71—21no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—41no—
189C4×C71—81no—
190C4×C71—81no—
191C4×C71—81no—
192#11698—711nomaximal, Sylow
193C2×C71—21no—
194C2×C71—21no—
195C2×C71—22no—
196C2×C71—22no—
197C2×C71—161no—
198C2×C71C2×#32711yescharacteristic
199C2×C71#212111yescharacteristic
200C2×C71#213311yescharacteristic
201#2618D7111yescharacteristic
202C2×#343—711no—
203C2×#324—711no—
204#1346—712no—
205#1345—712no—
206C71#1169811yescharacteristic, complemented, Sylow
207C22×M16—1421no—
208C2×#123—21no—
209C2×#121Dic7111yescharacteristic
210C22×C4⋊C4C2×D7111yescharacteristic
211C2×C22⋊C8—1421no—
212#398Dic7111yescharacteristic
213#398C2×D7111yescharacteristic
214#396—21no—
215#396—21no—
216#394Dic7111yescharacteristic
217#394C2×D7111yescharacteristic
218#386—21no—
219#380Dic7111yescharacteristic
220#343—712no—
221#343—1421no—
222#324—712no—
223#324—1421no—
224C22×D4—21no—
225C22×D4C2×Dic7111yescharacteristic
226C23×C4C2×Dic7111yescharacteristic
227C23×C4C22⋊D7111yescharacteristic
228C23×C4C22⋊D7111yescharacteristic
229C2×M16—2841no—
230C2×M16—2842no—
231C2×M16—1424no—
232C2×M16—1422no—
233C22×C8—2841no—
234C22×C8—2841no—
235#123—42no—
236#123—42no—
237#121—41no—
238#121—41no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C2×Dic7111yescharacteristic
245C2×C4⋊C4C2×Dic7111yescharacteristic
246C2×C4⋊C4C22⋊D7111yescharacteristic
247C2×C4⋊C4C22⋊D7111yescharacteristic
248C2×C4⋊C4C22×D7111yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C22⋊C8—2841no—
256C22⋊C8—1422no—
257#95—21no—
258#95—21no—
259#95C2×Dic7111yescharacteristic
260#95C2×Dic7111yescharacteristic
261C24—21no—
262C24C22⋊Dic7111yescharacteristic
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4C22⋊Dic7111yescharacteristic
282C22×C4C22⋊Dic7111yescharacteristic
283C22×C4C22⋊Dic7111yescharacteristic
284C22×C4#1176585012yes—
285C22×C4#1176585312yes—
286C22×C4C22×Dic7111yescharacteristic
287C22×C4C2×C22⋊D7111yescharacteristic
288C22×C4C2×C22⋊D7111yescharacteristic
289M16—2844no—
290M16—2842no—
291M16—1424no—
292C2×C8—2842no—
293C2×C8—2844no—
294C2×C8—2842no—
295C2×C8—5682no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#1455604311yescharacteristic
322C23#1455604311yescharacteristic
323C23#1455604912yes—
324C23#1455605512yes—
325C23C2×C22⋊Dic7111yescharacteristic
326D4—82no—
327D4—82no—
328D4—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4#1455604912yes—
359C2×C4C2×#1176585011yescharacteristic
360C2×C4C2×#1176585311yescharacteristic
361C8—2844no—
362C8—5682no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#1798573312yes—
379C22#1798573312yes—
380C22C2×#1455604311yescharacteristic
381C22C2×#1455604911yescharacteristic
382C22C2×#1455605511yescharacteristic
383C4—21no—
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—81no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—161no—
398C2C2×#1798573311yescharacteristic
399C2#2258998611yescharacteristic
400C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2618C2
socleC22×C71C2×#99
prime radicalC2×#121×C71C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2618D71
hypocenterC71#11698
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#17985768C2semidirect
2. C2×#17985725C2semidirect
3. #22586346C2semidirect
4. #22586343C2semidirect
206. C71#11698semidirect

Maximal subgroups

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

subgroupindex
C2×#179857682
C2×#179857252
#225863462
#225863432
C71×#26182
#1169871