Finite Group Databasebeta

← #34005208

Subgroups — #34005208

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—42no—
32#14556547—42no—
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×#11924007—21no—
38C2×#11924007—21no—
39C2×C22⋊Dic73—21no—
40C2×C4×Dic73—24no—
41C2×C4×Dic73—21no—
42C2×C4×Dic73—21no—
43C2×C4×Dic73C2×C411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73D411yescharacteristic
46C2×C4×Dic73C2311yescharacteristic
47#14556498—21no—
48#14556498—21no—
49#14556498—21no—
50C23×C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73—21no—
54#95×C73—21no—
55C2×#10596833—24no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—22no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—41no—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73D812yes—
71C22×Dic73SD1612yes—
72C22×Dic73C2×D411yescharacteristic
73C22×Dic73C2×D411yescharacteristic
74C4⋊Dic73—22no—
75C4⋊Dic73—22no—
76#11924007—42no—
77#11924007—42no—
78#11924007—42no—
79#11924007—42no—
80C22⋊Dic73—42no—
81C4×Dic73—22no—
82C4×Dic73—22no—
83C4×Dic73—22no—
84C4×Dic73—41no—
85C4×Dic73—41no—
86C4×Dic73—41no—
87C4×Dic73—41no—
88C4×Dic73#4114yes—
89C24×C73C22⋊C411yescharacteristic
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—22no—
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C22×C411yescharacteristic
97C4⋊C4×C73—22no—
98C4⋊C4×C73—22no—
99C22⋊C4×C73—42no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—21no—
105C23×C73—21no—
106C23×C73#10212yes—
107C23×C73C2×C22⋊C411yescharacteristic
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—44no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73#11712yes—
119C2×C4×C73#13112yes—
120C2×C4×C73#13112yes—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—21no—
124C2×Dic73—21no—
125C2×Dic73—21no—
126C2×Dic73—21no—
127C2×Dic73—24no—
128C2×Dic73—24no—
129C2×Dic73—44no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73#10212yes—
137C2×Dic73#10412yes—
138C2×Dic73#12312yes—
139C2×Dic73#12312yes—
140C2×Dic73C2×D811yescharacteristic
141C2×Dic73C2×SD1611yescharacteristic
142C2×Dic73#13712yes—
143#10596833—44no—
144#10596833—82no—
145C22×C73—22no—
146C22×C73—22no—
147C22×C73—22no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73#38111yescharacteristic
153C22×C73#40811yescharacteristic
154C22×C73#40811yescharacteristic
155C22×C73C2×#10211yescharacteristic
156C22×C73#41911yescharacteristic
157C22×C73C2×#10411yescharacteristic
158C22×C73#42111yescharacteristic
159C4×C73—41no—
160C4×C73—81no—
161C4×C73—81no—
162C4×C73#43314yes—
163Dic73—22no—
164Dic73—22no—
165Dic73—22no—
166Dic73—81no—
167Dic73—81no—
168Dic73#48012yes—
169Dic73#48312yes—
170#8709—731nomaximal, Sylow
171C2×C73—42no—
172C2×C73#148011yescharacteristic
173C2×C73#148511yescharacteristic
174C2×C73#149211yescharacteristic
175C2×C73#149711yescharacteristic
176C2×C73#181811yescharacteristic
177C2×C73#182311yescharacteristic
178C2×C73#182611yescharacteristic
179#2285—731no—
180#1797—731no—
181C4×C22⋊C8—731no—
182#1279—732no—
183#1279—732no—
184C73#870911yescharacteristic, complemented, Sylow
185C2×#117—1461no—
186C22×C42—731no—
187C2×C4⋊C8—732no—
188C2×C22⋊C8—1461no—
189C2×C22⋊C8—1461no—
190C2×C4×C8—732no—
191#384—731no—
192#384—731no—
193#382—1461no—
194#380—731no—
195#380—1461no—
196#380#6440811yescharacteristic
197C4×C4⋊C4—731no—
198C4×C4⋊C4—731no—
199C4×C4⋊C4—1461no—
200#336—1461no—
201#336—1461no—
202C23×C4—731no—
203C23×C4—731no—
204C23×C4C2×#6440811yescharacteristic
205C22×C8—1462no—
206C22×C8—1462no—
207#117—2922no—
208#117—2922no—
209C2×C4⋊C4—731no—
210C2×C4⋊C4—731no—
211C2×C4⋊C4—1461no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C4⋊C4C2×#6440811yescharacteristic
216C2×C4⋊C4C2×#6440811yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—1461no—
219C2×C42—731no—
220C2×C42—731no—
221C2×C42—731no—
222C2×C42—731no—
223C2×C42—1461no—
224C2×C42—1461no—
225C2×C42—1464no—
226C4⋊C8—1464no—
227C22⋊C8—2922no—
228C22⋊C8—1464no—
229C4×C8—1464no—
230#95—21no—
231#95—1461no—
232#95—1461no—
233#95—1461no—
234C24C22⋊#6440811yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—2921no—
241C22×C4—731no—
242C22×C4—732no—
243C22×C4—732no—
244C22×C4—731no—
245C22×C4—731no—
246C22×C4—731no—
247C22×C4—1461no—
248C22×C4—1461no—
249C22×C4—1462no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4—1462no—
253C22×C4C22⋊#6440811yescharacteristic
254C22×C4C22×#6440811yescharacteristic
255C2×C8—2922no—
256C2×C8—2922no—
257C2×C8—2922no—
258C2×C8—2922no—
259C2×C8—2922no—
260C2×C8—1464no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—2922no—
264C4⋊C4—2922no—
265C4⋊C4—2922no—
266C4⋊C4—2922no—
267C4⋊C4—1462no—
268C4⋊C4—1462no—
269C22⋊C4—42no—
270C22⋊C4—2922no—
271C42—2921no—
272C42—2921no—
273C42—2921no—
274C42—2921no—
275C42—734no—
276C42—1462no—
277C42—1462no—
278C42—1462no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#1455665312yes—
286C23C2×C22⋊#6440811yescharacteristic
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2924no—
304C2×C4—732no—
305C2×C4—731no—
306C2×C4—731no—
307C2×C4—732no—
308C2×C4—732no—
309C2×C4—732no—
310C2×C4—732no—
311C2×C4—1464no—
312C2×C4—1464no—
313C2×C4—1461no—
314C2×C4—1461no—
315C2×C4—1461no—
316C2×C4—1461no—
317C2×C4—1461no—
318C2×C4—1461no—
319C2×C4#1455667012yes—
320C2×C4#1455667812yes—
321C2×C4#1455667812yes—
322C8—2924no—
323C8—5842no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#1798953011yescharacteristic
332C22#1798956011yescharacteristic
333C22#1798956011yescharacteristic
334C22C2×#1455665311yescharacteristic
335C22#1798958011yescharacteristic
336C22D73⋊#10411yescharacteristic
337C22#1798958411yescharacteristic
338C4—41no—
339C4—81no—
340C4—81no—
341C4—5841no—
342C4—5841no—
343C4—732no—
344C4—732no—
345C4—1462no—
346C4—1462no—
347C4—1462no—
348C4#1798961114yes—
349C2—42no—
350C2#2262523611yescharacteristic
351C2#2262525311yescharacteristic
352C2#2262527611yescharacteristic
353C2#2262529311yescharacteristic
354C2#2262582411yescharacteristic
355C2#2262583411yescharacteristic
356C2#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#8709
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
184. C73#8709semidirect

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
#870973