Finite Group Databasebeta

← #34015962

Subgroups — #34015962

Order 18688.

1214 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626607C212yesmaximal
2#22625194C214yesmaximal, complemented
3#22625179C214yesmaximal
4#22625174C214yesmaximal, complemented
5C22×Q8×Dic73C211yesmaximal, characteristic
6#17989608—22no—
7#17989607C2214yes—
8#17989604—22no—
9#17989603C2214yes—
10#17989590—22no—
11#17989557C2214yes—
12C4×#11924037—22no—
13#17989432C22116yes—
14C2×Q8×Dic73—22no—
15C2×Q8×Dic73—22no—
16C2×Q8×Dic73—22no—
17C2×Q8×Dic73—22no—
18C2×Q8×Dic73C412yes—
19C2×Q8×Dic73C412yes—
20C2×Q8×Dic73C2212yes—
21C2×Q8×Dic73C2212yes—
22C22×C4⋊Dic73C411yescharacteristic
23C22×C4⋊Dic73C2212yes—
24C22×C4×Dic73C412yes—
25C22×C4×Dic73C2211yescharacteristic
26C23×Q8×C73C411yescharacteristic
27C2×#11924037—24no—
28#14556658—24no—
29C4⋊#10596833—24no—
30C22⋊#10596833—24no—
31C22⋊#10596833C2318yes—
32#14556646—24no—
33C4×#10596833—24no—
34C23×Dic73C2×C411yescharacteristic
35C23×Dic73C2×C412yes—
36C23×Dic73C2311yescharacteristic
37Q8×Dic73—24no—
38Q8×Dic73—24no—
39Q8×Dic73—24no—
40Q8×Dic73—24no—
41Q8×Dic73—24no—
42Q8×Dic73—24no—
43Q8×Dic73—24no—
44Q8×Dic73—24no—
45C2×C4⋊Dic73—24no—
46C2×C4⋊Dic73—24no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73C2×C412yes—
50C2×C4⋊Dic73C2×C412yes—
51C2×C4⋊Dic73C2×C414yes—
52C2×C4⋊Dic73C2314yes—
53C2×C4×Dic73—22no—
54C2×C4×Dic73—22no—
55C2×C4×Dic73—22no—
56C2×C4×Dic73—22no—
57C2×C4×Dic73C2×C414yes—
58C2×C4×Dic73C2×C414yes—
59C2×C4×Dic73C2×C411yescharacteristic
60C2×C4×Dic73C2×C411yescharacteristic
61C2×C4×Dic73D412yes—
62C2×C4×Dic73D412yes—
63C2×C4×Dic73D412yes—
64C2×C4×Dic73D412yes—
65C2×C4×Dic73C2311yescharacteristic
66C2×C4×Dic73C2311yescharacteristic
67C22×Q8×C73—28no—
68C22×Q8×C73—22no—
69C22×Q8×C73—22no—
70C22×Q8×C73C2×C412yes—
71C22×Q8×C73C2×C412yes—
72C23×C4×C73C2×C412yes—
73C23×C4×C73C2×C411yescharacteristic
74#11924037—48no—
75C2×#10596833—28no—
76C22×Dic73—24no—
77C22×Dic73—22no—
78C22×Dic73—22no—
79C22×Dic73—42no—
80C22×Dic73—44no—
81C22×Dic73C22⋊C411yescharacteristic
82C22×Dic73C22⋊C411yescharacteristic
83C22×Dic73C22⋊C411yescharacteristic
84C22×Dic73C22⋊C411yescharacteristic
85C22×Dic73C22×C411yescharacteristic
86C22×Dic73C22×C411yescharacteristic
87C22×Dic73C22×C414yes—
88C22×Dic73C22×C411yescharacteristic
89C22×Dic73C2×D412yes—
90C22×Dic73C2×D412yes—
91C22×Dic73C2411yescharacteristic
92C4⋊Dic73—24no—
93C4⋊Dic73—24no—
94C4⋊Dic73—28no—
95C4⋊Dic73—28no—
96C4×Dic73—24no—
97C4×Dic73—24no—
98C4×Dic73—24no—
99C4×Dic73—24no—
100C4×Dic73C22⋊C412yes—
101C4×Dic73C22⋊C412yes—
102C4×Dic73C22⋊C412yes—
103C4×Dic73C22⋊C412yes—
104C4×Dic73C2×D412yes—
105C4×Dic73C2×D412yes—
106C4×Dic73C2×D412yes—
107C4×Dic73C2×D412yes—
108C24×C73C22×C411yescharacteristic
109C2×Q8×C73—216no—
110C2×Q8×C73—216no—
111C2×Q8×C73—24no—
112C2×Q8×C73—24no—
113C2×Q8×C73—28no—
114C2×Q8×C73—24no—
115C2×Q8×C73M1618yescomplemented
116C22×C4×C73—22no—
117C22×C4×C73—22no—
118C22×C4×C73—28no—
119C22×C4×C73—24no—
120C22×C4×C73C22⋊C412yes—
121C22×C4×C73C22⋊C412yes—
122C22×C4×C73C22×C411yescharacteristic
123C22×C4×C73C22×C414yes—
124C22×C4×C73C22×C411yescharacteristic
125C23×C73—22no—
126C23×C73—22no—
127C23×C73—22no—
128C23×C73C2×C22⋊C411yescharacteristic
129C23×C73C2×C22⋊C411yescharacteristic
130C23×C73C23×C411yescharacteristic
131Q8×C73—216no—
132Q8×C73—216no—
133C2×C4×C73—28no—
134C2×C4×C73—28no—
135C2×C4×C73—24no—
136C2×C4×C73—22no—
137C2×C4×C73—24no—
138C2×C4×C73—24no—
139C2×C4×C73—24no—
140C2×C4×C73C2×C22⋊C412yes—
141C2×C4×C73C2×C22⋊C412yes—
142C2×C4×C73C2×M1618yes—
143C2×C4×C73C2×M1614yes—
144C2×Dic73—24no—
145C2×Dic73—22no—
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—22no—
149C2×Dic73—22no—
150C2×Dic73—24no—
151C2×Dic73—41no—
152C2×Dic73—42no—
153C2×Dic73—41no—
154C2×Dic73—44no—
155C2×Dic73—44no—
156C2×Dic73—44no—
157C2×Dic73#10114yes—
158C2×Dic73C2×C22⋊C411yescharacteristic
159C2×Dic73C2×C22⋊C411yescharacteristic
160C2×Dic73C2×C22⋊C412yes—
161C2×Dic73C2×C22⋊C411yescharacteristic
162C2×Dic73C2×C22⋊C411yescharacteristic
163C2×Dic73C22×D412yes—
164#10596833—48no—
165C22×C73—28no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73#41012yes—
173C22×C73C2×#10112yes—
174C22×C73C22×C22⋊C411yescharacteristic
175C22×C73C22×M1612yes—
176C4×C73—22no—
177C4×C73—28no—
178C4×C73#40714yes—
179Dic73—22no—
180Dic73—22no—
181Dic73—21no—
182Dic73—21no—
183Dic73—44no—
184Dic73C2×#10112yes—
185Dic73C2×#10112yes—
186#17365—731nomaximal, Sylow
187C2×C73—22no—
188C2×C73—22no—
189C2×C73—22no—
190C2×C73C2×#40711yescharacteristic
191C2×C73C2×#41011yescharacteristic
192C2×C73C22×#10111yescharacteristic
193C22×C4×Q8—731no—
194#2920—732no—
195#1467—734no—
196#1463—734no—
197#1461—734no—
198C73#1736511yescharacteristic, complemented, Sylow
199C23×Q8#6440811yescharacteristic
200C2×C4×Q8—732no—
201C2×C4×Q8—732no—
202C2×C4×Q8—732no—
203C2×C4×Q8—732no—
204C2×C4×Q8—1462no—
205C2×C4×Q8—1462no—
206C2×C4×Q8—1462no—
207C2×C4×Q8—1462no—
208C22×C4⋊C4—732no—
209C22×C4⋊C4—731no—
210C22×C42—731no—
211C22×C42—732no—
212#432—734no—
213#432—1462no—
214#431—734no—
215#431—1462no—
216#423—1462no—
217#407—734no—
218C4×M16—1462no—
219#324—7316no—
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8C2×#6440812yes—
224C22×Q8C2×#6440812yes—
225C23×C4—731no—
226C23×C4—732no—
227C23×C4—731no—
228C23×C4C2×#6440812yes—
229C23×C4C2×#6440811yescharacteristic
230C2×M16—1464no—
231C4×Q8—1464no—
232C4×Q8—1464no—
233C4×Q8—1464no—
234C4×Q8—1464no—
235C4×Q8—1464no—
236C4×Q8—1464no—
237C4×Q8—1464no—
238C4×Q8—1464no—
239C2×C4⋊C4—734no—
240C2×C4⋊C4—734no—
241C2×C4⋊C4—732no—
242C2×C4⋊C4—732no—
243C2×C4⋊C4—1464no—
244C2×C4⋊C4—1464no—
245C2×C4⋊C4—1462no—
246C2×C4⋊C4—1462no—
247C2×C42—732no—
248C2×C42—732no—
249C2×C42—732no—
250C2×C42—732no—
251C2×C42—731no—
252C2×C42—731no—
253C2×C42—731no—
254C2×C42—731no—
255C2×C42—734no—
256C2×C42—734no—
257C2×C42—1462no—
258C2×C42—1462no—
259C2×C42—1462no—
260C2×C42—1462no—
261C4⋊C8—1464no—
262C4⋊C8—1464no—
263C22⋊C8—738no—
264C22⋊C8—1464no—
265#97—1464no—
266C4×C8—1464no—
267C24C22×#6440811yescharacteristic
268C2×Q8—216no—
269C2×Q8—216no—
270C2×Q8—28no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8#1192403718yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—2924no—
280C22×C4—2922no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—732no—
284C22×C4—734no—
285C22×C4—731no—
286C22×C4—731no—
287C22×C4—731no—
288C22×C4—731no—
289C22×C4—731no—
290C22×C4—731no—
291C22×C4—732no—
292C22×C4—1462no—
293C22×C4—1462no—
294C22×C4—1464no—
295C22×C4C22⋊#6440812yes—
296C22×C4C22⋊#6440812yes—
297C22×C4C22×#6440814yes—
298C22×C4C22×#6440811yescharacteristic
299C22×C4C22×#6440811yescharacteristic
300M16—2928no—
301C2×C8—1468no—
302C4⋊C4—1468no—
303C4⋊C4—1468no—
304C4⋊C4—1464no—
305C4⋊C4—1464no—
306C42—732no—
307C42—732no—
308C42—732no—
309C42—732no—
310C42—732no—
311C42—732no—
312C42—732no—
313C42—732no—
314C42—1464no—
315C42—1464no—
316C42—1464no—
317C42—1464no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23C2×C22⋊#6440811yescharacteristic
322C23C2×C22⋊#6440811yescharacteristic
323C23C23×#6440811yescharacteristic
324Q8—216no—
325Q8—216no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—2924no—
334C2×C4—2924no—
335C2×C4—2924no—
336C2×C4—2921no—
337C2×C4—2922no—
338C2×C4—2921no—
339C2×C4—731no—
340C2×C4—731no—
341C2×C4—734no—
342C2×C4—731no—
343C2×C4—731no—
344C2×C4—732no—
345C2×C4—732no—
346C2×C4—1464no—
347C2×C4—1462no—
348C2×C4—1462no—
349C2×C4—1462no—
350C2×C4—1462no—
351C2×C4—1464no—
352C2×C4—1462no—
353C2×C4C2×C22⋊#6440812yes—
354C2×C4C2×C22⋊#6440812yes—
355C2×C4C2×#1192403718yes—
356C2×C4C2×#1192403714yes—
357C8—2928no—
358C22—28no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#1798956812yes—
366C22D73⋊#10112yes—
367C22C22×C22⋊#6440811yescharacteristic
368C22C22×#1192403712yes—
369C4—28no—
370C4—22no—
371C4—2924no—
372C4—732no—
373C4—732no—
374C4—1462no—
375C4—1462no—
376C4—1461no—
377C4—1461no—
378C4#1798955714yes—
379C2—22no—
380C2—22no—
381C2—22no—
382C2C2×#1798955711yescharacteristic
383C2C2×#1798956811yescharacteristic
384C2C2×D73⋊#10111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC23×Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenterC23×Q8#64408
hypocenterC73#17365
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #22625194C2semidirect
4. #22625174C2semidirect
115. C2×Q8×C73M16semidirect
198. C73#17365semidirect
274. C2×Q8#11924037semidirect

Maximal subgroups

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

subgroupindex
#226266072
#226251942
#226251792
#226251742
C22×Q8×Dic732
#1736573