Finite Group Databasebeta

← #34005211

Subgroups — #34005211

Order 18688.

540 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 124 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#22612333C211yesmaximal, characteristic
6C2×C4⋊#10596833C2212yes—
7C2×C22⋊#10596833—21no—
8C2×C22⋊#10596833—21no—
9C2×C4×#10596833C2212yes—
10#17989453—21no—
11#17989453—21no—
12C2×#14556547—21no—
13C22×C4×Dic73C2211yescharacteristic
14#17988456C411yescharacteristic
15#17988456C411yescharacteristic
16#17988451—21no—
17#17988447C411yescharacteristic
18#17988445—21no—
19C4⋊C4×Dic73C2211yescharacteristic
20C4⋊C4×Dic73C2211yescharacteristic
21#17988438—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
31#14556547—44no—
32C2×C4⋊Dic73C2×C411yescharacteristic
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×#11924007—21no—
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×#11924007—21no—
38C2×C22⋊Dic73—21no—
39C2×C4×Dic73—24no—
40C2×C4×Dic73—21no—
41C2×C4×Dic73—21no—
42C2×C4×Dic73C2×C411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73C2311yescharacteristic
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C23×C4×C73C2×C411yescharacteristic
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73—21no—
53#95×C73—21no—
54C2×#10596833—24no—
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C22×Dic73—22no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—41no—
67C22×Dic73C22⋊C411yescharacteristic
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73SD1612yes—
70C22×Dic73Q1612yes—
71C22×Dic73C2×D411yescharacteristic
72C22×Dic73C2×D411yescharacteristic
73C4⋊Dic73—22no—
74C4⋊Dic73—22no—
75#11924007—42no—
76#11924007—42no—
77#11924007—42no—
78#11924007—42no—
79C22⋊Dic73—42no—
80C4×Dic73—22no—
81C4×Dic73—22no—
82C4×Dic73—22no—
83C4×Dic73—42no—
84C4×Dic73—42no—
85C4×Dic73#4114yes—
86C24×C73C22⋊C411yescharacteristic
87C22×C4×C73—22no—
88C22×C4×C73—22no—
89C22×C4×C73—22no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22×C411yescharacteristic
94C4⋊C4×C73—22no—
95C4⋊C4×C73—22no—
96C22⋊C4×C73—42no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73#10312yes—
104C23×C73C2×C22⋊C411yescharacteristic
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—44no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73#11712yes—
116C2×C4×C73#13112yes—
117C2×C4×C73#13112yes—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—24no—
125C2×Dic73—24no—
126C2×Dic73—44no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73#10312yes—
134C2×Dic73#10412yes—
135C2×Dic73#12312yes—
136C2×Dic73#12312yes—
137C2×Dic73C2×SD1611yescharacteristic
138C2×Dic73C2×Q1611yescharacteristic
139C2×Dic73AGL(1,ℤ/8)12yes—
140#10596833—44no—
141#10596833—82no—
142C22×C73—22no—
143C22×C73—22no—
144C22×C73—22no—
145C22×C73—41no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73#38111yescharacteristic
150C22×C73#40811yescharacteristic
151C22×C73#40811yescharacteristic
152C22×C73C2×#10311yescharacteristic
153C22×C73#41811yescharacteristic
154C22×C73C2×#10411yescharacteristic
155C22×C73#42111yescharacteristic
156C4×C73—41no—
157C4×C73—81no—
158C4×C73—81no—
159C4×C73#43314yes—
160Dic73—22no—
161Dic73—22no—
162Dic73—22no—
163Dic73—81no—
164Dic73—81no—
165Dic73#48112yes—
166Dic73#48412yes—
167#8710—731nomaximal, Sylow
168C2×C73—42no—
169C2×C73#148111yescharacteristic
170C2×C73#148411yescharacteristic
171C2×C73#149311yescharacteristic
172C2×C73#149611yescharacteristic
173C2×C73#181811yescharacteristic
174C2×C73#182411yescharacteristic
175C2×C73#182611yescharacteristic
176#2285—731no—
177#1797—731no—
178C4×C22⋊C8—731no—
179#1279—732no—
180#1279—732no—
181C73#871011yescharacteristic, complemented, Sylow
182C2×#117—1461no—
183C22×C42—731no—
184C2×C4⋊C8—732no—
185C2×C22⋊C8—1461no—
186C2×C22⋊C8—1461no—
187C2×C4×C8—732no—
188#384—731no—
189#384—731no—
190#382—1461no—
191#380—731no—
192#380—1461no—
193#380#6440811yescharacteristic
194C4×C4⋊C4—731no—
195C4×C4⋊C4—731no—
196C4×C4⋊C4—1461no—
197#336—1461no—
198#336—1461no—
199C23×C4—731no—
200C23×C4—731no—
201C23×C4C2×#6440811yescharacteristic
202C22×C8—1462no—
203C22×C8—1462no—
204#117—2924no—
205C2×C4⋊C4—731no—
206C2×C4⋊C4—731no—
207C2×C4⋊C4—1461no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4—1461no—
211C2×C4⋊C4C2×#6440811yescharacteristic
212C2×C4⋊C4C2×#6440811yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1461no—
215C2×C42—731no—
216C2×C42—731no—
217C2×C42—731no—
218C2×C42—731no—
219C2×C42—1461no—
220C2×C42—1461no—
221C2×C42—1464no—
222C4⋊C8—1464no—
223C22⋊C8—2922no—
224C22⋊C8—1464no—
225C4×C8—1464no—
226#95—21no—
227#95—1461no—
228#95—1461no—
229#95—1461no—
230C24C22⋊#6440811yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—2921no—
237C22×C4—731no—
238C22×C4—732no—
239C22×C4—732no—
240C22×C4—731no—
241C22×C4—731no—
242C22×C4—731no—
243C22×C4—1461no—
244C22×C4—1461no—
245C22×C4—1462no—
246C22×C4—1462no—
247C22×C4—1462no—
248C22×C4—1462no—
249C22×C4C22⋊#6440811yescharacteristic
250C22×C4C22×#6440811yescharacteristic
251C2×C8—2922no—
252C2×C8—2922no—
253C2×C8—2922no—
254C2×C8—2922no—
255C2×C8—2922no—
256C2×C8—1464no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—2922no—
260C4⋊C4—2922no—
261C4⋊C4—2922no—
262C4⋊C4—2922no—
263C4⋊C4—1462no—
264C4⋊C4—1462no—
265C22⋊C4—42no—
266C22⋊C4—2922no—
267C42—2922no—
268C42—2922no—
269C42—734no—
270C42—1462no—
271C42—1462no—
272C42—1462no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1455665412yes—
280C23C2×C22⋊#6440811yescharacteristic
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2921no—
295C2×C4—2921no—
296C2×C4—2921no—
297C2×C4—2924no—
298C2×C4—732no—
299C2×C4—731no—
300C2×C4—731no—
301C2×C4—732no—
302C2×C4—732no—
303C2×C4—732no—
304C2×C4—732no—
305C2×C4—1464no—
306C2×C4—1464no—
307C2×C4—1461no—
308C2×C4—1461no—
309C2×C4—1461no—
310C2×C4—1461no—
311C2×C4—1461no—
312C2×C4—1461no—
313C2×C4#1455667012yes—
314C2×C4#1455667812yes—
315C2×C4#1455667812yes—
316C8—2924no—
317C8—5842no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#1798953011yescharacteristic
326C22#1798956011yescharacteristic
327C22#1798956011yescharacteristic
328C22C2×#1455665411yescharacteristic
329C22#1798957911yescharacteristic
330C22D73⋊#10411yescharacteristic
331C22#1798958411yescharacteristic
332C4—41no—
333C4—81no—
334C4—81no—
335C4—5841no—
336C4—5841no—
337C4—732no—
338C4—732no—
339C4—1462no—
340C4—1462no—
341C4—1462no—
342C4#1798961114yes—
343C2—42no—
344C2#2262523911yescharacteristic
345C2#2262525011yescharacteristic
346C2#2262527911yescharacteristic
347C2#2262529011yescharacteristic
348C2#2262582411yescharacteristic
349C2#2262583511yescharacteristic
350C2#2262583911yescharacteristic
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#8710
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
181. C73#8710semidirect

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
#226123332
#871073