Finite Group Databasebeta

← #34005214

Subgroups — #34005214

Order 18688.

496 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 116 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
1#22625785C211yesmaximal, characteristic
2C4×C22⋊#10596833C211yesmaximal, characteristic
3#22624767C212yesmaximal, complemented
4#22624767C212yesmaximal, complemented
5#22612326C211yesmaximal, characteristic
6C2×C4⋊#10596833C2212yes—
7C2×C22⋊#10596833—21no—
8C2×C22⋊#10596833—21no—
9C2×C4×#10596833C2212yes—
10#17989453—21no—
11#17989453—21no—
12C22×C4×Dic73C2211yescharacteristic
13#17988454—21no—
14#17988452C411yescharacteristic
15#17988452C411yescharacteristic
16#17988448—21no—
17#17988447C411yescharacteristic
18C4⋊C4×Dic73C2211yescharacteristic
19C4⋊C4×Dic73C2211yescharacteristic
20C22⋊C4×Dic73—21no—
21#17988432—21no—
22#380×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25C4⋊#10596833—24no—
26C22⋊#10596833—24no—
27C22⋊#10596833—42no—
28C4×#10596833—24no—
29C23×Dic73D411yescharacteristic
30C23×Dic73D411yescharacteristic
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×C4⋊Dic73C2×C411yescharacteristic
33C2×C22⋊Dic73—21no—
34C2×C4×Dic73—24no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—21no—
37C2×C4×Dic73C2×C411yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73C2311yescharacteristic
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48C23×C4×C73C2×C411yescharacteristic
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73—21no—
52#95×C73—21no—
53C2×#10596833—24no—
54C2×#10596833—42no—
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C22×Dic73—22no—
60C22×Dic73—22no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—21no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—41no—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73C2×D411yescharacteristic
71C22×Dic73C2×D411yescharacteristic
72C4⋊Dic73—22no—
73C4⋊Dic73—22no—
74C22⋊Dic73—42no—
75C4×Dic73—22no—
76C4×Dic73—22no—
77C4×Dic73—22no—
78C4×Dic73—42no—
79C4×Dic73—41no—
80C4×Dic73—41no—
81C4×Dic73#4114yes—
82C24×C73C22⋊C411yescharacteristic
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22×C411yescharacteristic
90C4⋊C4×C73—22no—
91C4⋊C4×C73—22no—
92C22⋊C4×C73—42no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73#10412yes—
100C23×C73C2×C22⋊C411yescharacteristic
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—44no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73#11712yes—
112C2×C4×C73#13112yes—
113C2×C4×C73#13112yes—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—44no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73#12312yes—
132C2×Dic73#12312yes—
133C2×Dic73#13512yes—
134C2×Dic73AGL(1,ℤ/8)11yescharacteristic
135C2×Dic73#13711yescharacteristic
136#10596833—44no—
137#10596833—82no—
138C22×C73—22no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73#38111yescharacteristic
146C22×C73#40811yescharacteristic
147C22×C73#40811yescharacteristic
148C22×C73#41611yescharacteristic
149C22×C73#41711yescharacteristic
150C22×C73C2×#10411yescharacteristic
151C22×C73#42111yescharacteristic
152C4×C73—41no—
153C4×C73—81no—
154C4×C73—81no—
155C4×C73#43314yes—
156Dic73—22no—
157Dic73—41no—
158Dic73—41no—
159Dic73—81no—
160Dic73—81no—
161Dic73#48212yes—
162Dic73#48512yes—
163#8711—731nomaximal, Sylow
164C2×C73—42no—
165C2×C73#148211yescharacteristic
166C2×C73#148311yescharacteristic
167C2×C73#149411yescharacteristic
168C2×C73#149511yescharacteristic
169C2×C73#181811yescharacteristic
170C2×C73#182511yescharacteristic
171C2×C73#182611yescharacteristic
172#2283—731no—
173#1797—731no—
174C4×C22⋊C8—731no—
175#1279—732no—
176#1279—732no—
177C73#871111yescharacteristic, complemented, Sylow
178C22×C42—731no—
179C2×C4⋊C8—732no—
180C2×C22⋊C8—1461no—
181C2×C22⋊C8—1461no—
182C2×C4×C8—732no—
183#383—1461no—
184#382—731no—
185#382—731no—
186#381—1461no—
187#380—731no—
188#380#6440811yescharacteristic
189C4×C4⋊C4—731no—
190C4×C4⋊C4—731no—
191C4×C22⋊C4—1461no—
192#376—1461no—
193#336—1461no—
194#336—1461no—
195C23×C4—731no—
196C23×C4—731no—
197C23×C4C2×#6440811yescharacteristic
198C22×C8—1462no—
199C22×C8—1462no—
200C2×C4⋊C4—731no—
201C2×C4⋊C4—731no—
202C2×C4⋊C4C2×#6440811yescharacteristic
203C2×C4⋊C4C2×#6440811yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—1461no—
206C2×C42—731no—
207C2×C42—731no—
208C2×C42—731no—
209C2×C42—731no—
210C2×C42—1461no—
211C2×C42—1461no—
212C2×C42—1464no—
213C4⋊C8—1464no—
214C22⋊C8—2922no—
215C22⋊C8—1464no—
216C4×C8—1464no—
217#95—21no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222#95—1461no—
223#95—1461no—
224#95—1461no—
225C24C22⋊#6440811yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—2921no—
232C22×C4—731no—
233C22×C4—731no—
234C22×C4—731no—
235C22×C4—731no—
236C22×C4—1461no—
237C22×C4—1461no—
238C22×C4—1461no—
239C22×C4—1462no—
240C22×C4—1462no—
241C22×C4—1462no—
242C22×C4—1461no—
243C22×C4—1462no—
244C22×C4C22⋊#6440811yescharacteristic
245C22×C4C22×#6440811yescharacteristic
246C2×C8—2922no—
247C2×C8—2922no—
248C2×C8—2922no—
249C2×C8—2922no—
250C2×C8—2922no—
251C2×C8—1464no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—1462no—
255C4⋊C4—1462no—
256C22⋊C4—42no—
257C22⋊C4—2922no—
258C42—2922no—
259C42—2921no—
260C42—2921no—
261C42—734no—
262C42—1462no—
263C42—1462no—
264C42—1462no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#1455665612yes—
272C23C2×C22⋊#6440811yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—2922no—
284C2×C4—2922no—
285C2×C4—2924no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—732no—
293C2×C4—732no—
294C2×C4—731no—
295C2×C4—731no—
296C2×C4—732no—
297C2×C4—1461no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1461no—
303C2×C4—1461no—
304C2×C4—1461no—
305C2×C4#1455667012yes—
306C2×C4#1455667812yes—
307C2×C4#1455667812yes—
308C8—2924no—
309C8—5842no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#1798953011yescharacteristic
318C22#1798956011yescharacteristic
319C22#1798956011yescharacteristic
320C22#1798957711yescharacteristic
321C22#1798957811yescharacteristic
322C22C2×#1455665611yescharacteristic
323C22#1798958511yescharacteristic
324C4—41no—
325C4—81no—
326C4—81no—
327C4—2921no—
328C4—2921no—
329C4—5841no—
330C4—5841no—
331C4—732no—
332C4—732no—
333C4—1462no—
334C4#1798961114yes—
335C2—42no—
336C2#2262524211yescharacteristic
337C2#2262524811yescharacteristic
338C2#2262528411yescharacteristic
339C2#2262528611yescharacteristic
340C2#2262582411yescharacteristic
341C2#2262583611yescharacteristic
342C2#2262583811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#380×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#380#64408
hypocenterC73#8711
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #22624767C2semidirect
4. #22624767C2semidirect
177. C73#8711semidirect

Maximal subgroups

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

subgroupindex
#226257852
C4×C22⋊#105968332
#226247672
#226123262
#871173