Finite Group Databasebeta

← #34033206

Subgroups — #34033206

Order 18688.

630 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 97 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#22627072C212yesmaximal
2#22626932C214yesmaximal, complemented
3C2×(C2×#64408)⋊Q8C211yesmaximal, characteristic
4C2×C22⋊#10596842—22no—
5(C2×#10596842)⋊C4—22no—
6(C2×#10596842)⋊C4C2214yes—
7C22×C4⋊#64408C411yescharacteristic
8C22×C4×#64408C2211yescharacteristic
9(C2×#64408)⋊Q8—22no—
10(C2×#64408)⋊Q8C412yes—
11(C2×#64408)⋊Q8C2212yes—
12C2×(C2×#64408)⋊C4—22no—
13C22×Q8×D73C411yescharacteristic
14C22×#10596842—24no—
15C22⋊#10596842—44no—
16C23×#64408—21no—
17C23×#64408D412yes—
18C2×C4⋊#64408—21no—
19C2×C4⋊#64408—42no—
20C2×C4⋊#64408C2×C412yes—
21C2×C4×#64408—42no—
22C2×C4×#64408C2×C411yescharacteristic
23C2×C4×#64408D412yes—
24C2×C4×#64408C2311yescharacteristic
25(C2×#64408)⋊C4—24no—
26(C2×#64408)⋊C4—24no—
27C22×Dic146—21no—
28C22×Dic146C811yescharacteristic, complemented
29C2×Q8×D73—21no—
30C2×Q8×D73—21no—
31C2×Q8×D73—42no—
32C2×Q8×D73—42no—
33C2×Q8×D73C812yescomplemented
34C2×Q8×D73C2×C412yes—
35C22×C4×D73—21no—
36C22×C4×D73C2×C411yescharacteristic
37C22×Q8×C73C811yescharacteristic, complemented
38C2×#10596842—44no—
39C2×#10596842—44no—
40C2×#10596842—44no—
41C22×#64408—24no—
42C22×#64408—24no—
43C22×#64408—22no—
44C22×#64408—42no—
45C22×#64408—44no—
46C22×#64408—42no—
47C22×#64408C22⋊C412yes—
48C22×#64408C2×D412yes—
49C4⋊#64408—42no—
50C4⋊#64408—42no—
51C4×#64408—41no—
52C4×#64408—42no—
53C4×#64408—41no—
54C23×D73C22⋊C411yescharacteristic
55C2×Dic146—21no—
56C2×Dic146—22no—
57C2×Dic146—22no—
58C2×Dic146—44no—
59C2×Dic146—42no—
60C2×Dic146C2×C812yes—
61Q8×D73—44no—
62Q8×D73—42no—
63Q8×D73—42no—
64Q8×D73—44no—
65Q8×D73—42no—
66Q8×D73—42no—
67C22×Dic73—21no—
68C22×Dic73M1611yescharacteristic
69C2×C4×D73—22no—
70C2×C4×D73—22no—
71C2×C4×D73—22no—
72C2×C4×D73—22no—
73C2×C4×D73—42no—
74C2×C4×D73—42no—
75C2×C4×D73C22⋊C411yescharacteristic
76C2×C4×D73C2×C811yescharacteristic
77C2×C4×D73M1611yescharacteristic
78C2×C4×D73C22×C411yescharacteristic
79C2×Q8×C73—21no—
80C2×Q8×C73—42no—
81C2×Q8×C73C2×C812yes—
82C22×C4×C73—21no—
83C22×C4×C73C2×C811yescharacteristic
84C23×C73C22⋊C811yescharacteristic
85C22×D73—22no—
86C22×D73—22no—
87C22×D73C22⋊C811yescharacteristic
88C22×D73#9912yes—
89C22×D73#10012yes—
90C22×D73C2×C22⋊C411yescharacteristic
91C2×#64408—22no—
92C2×#64408—22no—
93C2×#64408—22no—
94C2×#64408—22no—
95C2×#64408—22no—
96C2×#64408—48no—
97C2×#64408—42no—
98C2×#64408—44no—
99C2×#64408—84no—
100C2×#64408#13714yes—
101#10596842—84no—
102Q8×C73—42no—
103Q8×C73—42no—
104C2×C4×C73—22no—
105C2×C4×C73—22no—
106C2×C4×C73—42no—
107C2×C4×C73C22⋊C811yescharacteristic
108C2×C4×C73C22×C811yescharacteristic
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—42no—
112C2×Dic73C22⋊C811yescharacteristic
113C2×Dic73C2×M1611yescharacteristic
114C4×D73—22no—
115C4×D73—22no—
116C4×D73—21no—
117C4×D73—21no—
118C4×D73—21no—
119C4×D73—21no—
120C4×D73—44no—
121C4×D73—42no—
122C4×D73—42no—
123Dic146—44no—
124Dic146—44no—
125Dic146—42no—
126Dic146—42no—
127C22×C73—22no—
128C22×C73#32312yes—
129C22×C73C2×C22⋊C811yescharacteristic
130C2×D73—22no—
131C2×D73—22no—
132C2×D73—24no—
133C2×D73—22no—
134C2×D73#32312yes—
135C2×D73C2×#9911yescharacteristic
136C2×D73C2×#10011yescharacteristic
137C2×D73#41912yes—
138C2×D73#42112yes—
139#64408—24no—
140#64408—42no—
141#64408—82no—
142C4×C73—21no—
143C4×C73—21no—
144C4×C73—42no—
145Dic73—21no—
146Dic73—21no—
147Dic73—42no—
148#7595—731nomaximal, Sylow
149C2×C73—22no—
150C2×C73C2×#32311yescharacteristic
151C2×C73#147912yes—
152D73—22no—
153D73#151112yes—
154D73#153311yescharacteristic
155D73#155811yescharacteristic
156C2×#391—731no—
157#1817—732no—
158#1272—734no—
159C73#759511yescharacteristic, complemented, Sylow
160C23×Q8—731no—
161C22×C4⋊C4—731no—
162C22×C42—731no—
163C2×C22⋊C8—1462no—
164#391—732no—
165#391—732no—
166#391—1462no—
167C2×#95—1462no—
168#336—734no—
169#336—1462no—
170C22×Q8—2922no—
171C22×Q8—2922no—
172C22×Q8—732no—
173C22×Q8—732no—
174C22×Q8—731no—
175C22×Q8—1461no—
176C22×Q8—1461no—
177C22×Q8—1461no—
178C22×Q8#1059684211yescharacteristic, complemented
179C23×C4—732no—
180C23×C4—731no—
181C23×C4—1461no—
182C23×C4—1461no—
183C22×C8—1464no—
184C2×C4⋊C4—2922no—
185C2×C4⋊C4—732no—
186C2×C4⋊C4—1461no—
187C2×C42—2922no—
188C2×C42—731no—
189C2×C42—732no—
190C2×C42—731no—
191C22⋊C8—2924no—
192#95—1464no—
193#95—1464no—
194C24—731no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—2922no—
198C2×Q8—2924no—
199C2×Q8—2922no—
200C2×Q8—2922no—
201C2×Q8—2924no—
202C2×Q8—2922no—
203C2×Q8—2922no—
204C2×Q8—2924no—
205C2×Q8—732no—
206C2×Q8—1462no—
207C2×Q8—1462no—
208C2×Q8—1461no—
209C2×Q8C2×#1059684212yes—
210C22×C4—21no—
211C22×C4—2922no—
212C22×C4—2922no—
213C22×C4—2922no—
214C22×C4—2922no—
215C22×C4—2924no—
216C22×C4—732no—
217C22×C4—731no—
218C22×C4—731no—
219C22×C4—731no—
220C22×C4—731no—
221C22×C4—731no—
222C22×C4—732no—
223C22×C4—1462no—
224C22×C4—1462no—
225C22×C4—1462no—
226C22×C4—1462no—
227C22×C4—1462no—
228C22×C4—1464no—
229C22×C4—1464no—
230C22×C4—1461no—
231C22×C4C2×#1059684211yescharacteristic
232C2×C8—2924no—
233C2×C8—2924no—
234C2×C8—2924no—
235C4⋊C4—2922no—
236C4⋊C4—2922no—
237C42—2921no—
238C42—2922no—
239C42—2921no—
240C23—731no—
241C23—732no—
242C23—732no—
243C23—731no—
244C23—1462no—
245C23—1462no—
246C23C22⋊#1059684211yescharacteristic
247Q8—42no—
248Q8—42no—
249Q8—2922no—
250Q8—2922no—
251Q8—2924no—
252Q8—2924no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—2922no—
257C2×C4—2922no—
258C2×C4—2924no—
259C2×C4—2922no—
260C2×C4—2928no—
261C2×C4—2922no—
262C2×C4—2924no—
263C2×C4—5844no—
264C2×C4—734no—
265C2×C4—731no—
266C2×C4—731no—
267C2×C4—1461no—
268C2×C4—1461no—
269C2×C4—1461no—
270C2×C4—1461no—
271C2×C4—1462no—
272C2×C4—1462no—
273C2×C4—1462no—
274C2×C4—1462no—
275C2×C4—1462no—
276C2×C4—1462no—
277C2×C4—1462no—
278C2×C4—1462no—
279C2×C4—1462no—
280C2×C4C22⋊#1059684211yescharacteristic
281C2×C4C22×#1059684211yescharacteristic
282C8—5844no—
283C22—22no—
284C22—731no—
285C22—731no—
286C22—732no—
287C22—732no—
288C22—732no—
289C22—1464no—
290C22—1462no—
291C22—1462no—
292C22—1462no—
293C22#1798967212yes—
294C22C2×C22⋊#1059684211yescharacteristic
295C4—21no—
296C4—21no—
297C4—42no—
298C4—2922no—
299C4—2922no—
300C4—5842no—
301C4—1464no—
302C4—1461no—
303C4—1461no—
304C2—22no—
305C2—732no—
306C2—731no—
307C2—731no—
308C2—1462no—
309C2C2×#1798967211yescharacteristic
310C2#2262705212yes—
subgroupG/H
centerC22C2×C22⋊#10596842
commutatorC2×C4×C73C22×C8
FrattiniC2×C4C22×#10596842
FittingC22×Q8×C73C8
socleC22×C73C2×C22⋊C8
prime radicalC23×D73C22⋊C4
cosocleC2×C4×#64408C23
hypercenterC22×Q8#10596842
hypocenterC73#7595
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #22626932C2semidirect
28. C22×Dic146C8semidirect
33. C2×Q8×D73C8semidirect
37. C22×Q8×C73C8semidirect
159. C73#7595semidirect
178. C22×Q8#10596842semidirect

Maximal subgroups

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

subgroupindex
#226270722
#226269322
C2×(C2×#64408)⋊Q82
#759573