Finite Group Databasebeta

← C2×#22615314 #33257219

Subgroups — C2×#22615314 #33257219

Order 18688.

2574 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556583C211yesmaximal, characteristic
2C22×C4×Dic146C211yesmaximal, characteristic
3#22615314C2116yesmaximal, direct product factor
4C2×#17988519C212yesmaximal
5C2×#17988508C212yesmaximal
6C2×#17988486C212yesmaximal
7C2×#17988486C212yesmaximal
8C2×#17988485C214yesmaximal
9C2×C4×C4⋊C4×C73C211yesmaximal, characteristic
10C23×Dic146C2211yescharacteristic
11C2×#14556583—28no—
12C2×#14556583C2218yes—
13C2×C4×Dic146—28no—
14C2×C4×Dic146C2218yes—
15C22×C4⋊Dic73C2211yescharacteristic
16C22×C4⋊Dic73C2214yes—
17C22×C4⋊Dic73C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C22×C4⋊Dic73C2214yes—
20C22×#11924007C2212yes—
21C22×#11924007C2212yes—
22C22×#11924007C2212yes—
23C22×C4×Dic73C2212yes—
24#17988519C22116yes—
25#17988508C22116yes—
26#17988486C22116yes—
27#17988486C22116yes—
28#17988485C22132yes—
29C2×#14556498C2214yes—
30C2×#14556498C2214yes—
31C22×C4⋊C4×C73C2212yes—
32C22×C42×C73C2212yes—
33C22×C42×C73C2211yescharacteristic
34C4×C4⋊C4×C73C2218yes—
35C2×#95×C73C2212yes—
36C22×Dic146—28no—
37C22×Dic146—24no—
38C22×Dic146C2314yes—
39C23×Dic73C2312yes—
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2312yes—
43#14556583—232no—
44C4×Dic146—232no—
45C2×C4⋊Dic73—216no—
46C2×C4⋊Dic73—24no—
47C2×C4⋊Dic73—24no—
48C2×C4⋊Dic73—24no—
49C2×C4⋊Dic73—48no—
50C2×C4⋊Dic73C23116yes—
51C2×C4⋊Dic73C2314yes—
52C2×C4⋊Dic73C2314yes—
53C2×C4⋊Dic73C2314yes—
54C2×C4⋊Dic73C23116yes—
55C2×#11924007—28no—
56C2×#11924007—28no—
57C2×#11924007—28no—
58C2×#11924007C2318yes—
59C2×#11924007C2318yes—
60C2×#11924007C2318yes—
61C2×C4×Dic73—28no—
62C2×C4×Dic73C2318yes—
63#14556498C23116yes—
64#14556498C23116yes—
65C23×C4×C73C2311yescharacteristic
66C23×C4×C73C2311yescharacteristic
67C23×C4×C73C2314yes—
68C23×C4×C73C2311yescharacteristic
69C2×C4⋊C4×C73Q8116yescomplemented
70C2×C4⋊C4×C73C2318yes—
71C2×C42×C73—44no—
72C2×C42×C73D418yes—
73C2×C42×C73C2318yes—
74C2×C42×C73C2314yes—
75#95×C73C2318yes—
76C2×Dic146—232no—
77C2×Dic146—28no—
78C2×Dic146—216no—
79C22×Dic73—24no—
80C22×Dic73—24no—
81C22×Dic73—24no—
82C22×Dic73—24no—
83C22×Dic73—24no—
84C22×Dic73—28no—
85C22×Dic73—24no—
86C22×Dic73—24no—
87C22×Dic73—24no—
88C22×Dic73—24no—
89C22×Dic73#4118yes—
90C22×Dic73C2414yes—
91C22×Dic73C2414yes—
92C22×Dic73C2414yes—
93C22×Dic73C2414yes—
94C4⋊Dic73—28no—
95C4⋊Dic73—28no—
96C4⋊Dic73—28no—
97C4⋊Dic73—232no—
98C4⋊Dic73—416no—
99#11924007—216no—
100#11924007—216no—
101#11924007—216no—
102C4×Dic73—216no—
103C24×C73C2411yescharacteristic
104C22×C4×C73—28no—
105C22×C4×C73—24no—
106C22×C4×C73—22no—
107C22×C4×C73—28no—
108C22×C4×C73—22no—
109C22×C4×C73C2×D414yes—
110C22×C4×C73C2×D414yes—
111C22×C4×C73C2×D414yes—
112C22×C4×C73C2×Q8116yes—
113C22×C4×C73C2×Q818yes—
114C22×C4×C73C2412yes—
115C22×C4×C73C2412yes—
116C22×C4×C73C2418yes—
117C22×C4×C73C2412yes—
118C4⋊C4×C73C2×Q8132yescomplemented
119C42×C73—48no—
120C42×C73C2×D4116yes—
121C23×C73C22×D412yes—
122C23×C73C22×Q814yes—
123C23×C73C2×#4114yes—
124C23×C73H(5,𝔽2)12yes—
125C23×C732−1+412yes—
126C23×C73C2511yescharacteristic
127C2×C4×C73—28no—
128C2×C4×C73—28no—
129C2×C4×C73—232no—
130C2×C4×C73—28no—
131C2×C4×C73—24no—
132C2×C4×C73—22no—
133C2×C4×C73—28no—
134C2×C4×C73—22no—
135C2×C4×C73C4⋊Q8116yes—
136C2×C4×C73C22×D414yes—
137C2×C4×C73C22×D414yes—
138C2×C4×C73C22×D414yes—
139C2×C4×C73C22×Q8116yes—
140C2×C4×C73C22×Q818yes—
141C2×Dic73—216no—
142C2×Dic73—24no—
143C2×Dic73—24no—
144C2×Dic73—24no—
145C2×Dic73—24no—
146C2×Dic73—24no—
147C2×Dic73—28no—
148C2×Dic73—24no—
149C2×Dic73—24no—
150C2×Dic73—24no—
151C2×Dic73—24no—
152C2×Dic73—48no—
153C2×Dic73—48no—
154C2×Dic73—48no—
155C2×Dic73C2×#4118yes—
156Dic146—232no—
157C22×C73#52514yes—
158C22×C73C2×C4⋊Q814yes—
159C22×C73#53714yes—
160C22×C73#55418yes—
161C22×C73#55718yes—
162C22×C73C23×D411yescharacteristic
163C22×C73C23×Q812yes—
164C22×C73C22×#4112yes—
165C22×C73C2×H(5,𝔽2)11yescharacteristic
166C22×C73C2×2−1+411yescharacteristic
167C4×C73—24no—
168C4×C73—24no—
169C4×C73—216no—
170C4×C73C2×C4⋊Q818yes—
171Dic73—28no—
172Dic73—44no—
173Dic73—44no—
174Dic73—44no—
175C2×#2584—731nomaximal, Sylow
176C2×C73#258418yescomplemented
177C2×C73C2×#52511yescharacteristic
178C2×C73C22×C4⋊Q811yescharacteristic
179C2×C73C2×#53711yescharacteristic
180C2×C73C2×#55412yes—
181C2×C73C2×#55712yes—
182C22×#125—731no—
183C22×C4×Q8—731no—
184#2584—7316no—
185C2×#398—732no—
186C2×#395—732no—
187C2×#389—734no—
188C2×#389—732no—
189C2×#389—732no—
190C2×C4×C4⋊C4D7311yescharacteristic
191C73C2×#258411yescharacteristic, complemented, Sylow
192C23×Q8—731no—
193C2×#125—738no—
194C2×#125—1468no—
195C2×C4×Q8—738no—
196C2×C4×Q8—1468no—
197C22×C4⋊C4—734no—
198C22×C4⋊C4—732no—
199C22×C4⋊C4—734no—
200C22×C4⋊C4—731no—
201C22×C4⋊C4—732no—
202C22×C4⋊C4—732no—
203C22×C4⋊C4—731no—
204C22×C4⋊C4—731no—
205C22×C4⋊C4C2×D7312yes—
206C22×C42—732no—
207C22×C42C2×D7312yes—
208C22×C42C2×D7311yescharacteristic
209#398—7316no—
210#395—7316no—
211#389—7316no—
212#389—7316no—
213#389—7332no—
214C4×C4⋊C4C2×D7318yes—
215C2×#95—734no—
216C2×#95—734no—
217C2×#95C2×D7312yes—
218C22×Q8—734no—
219C22×Q8—1468no—
220C22×Q8—1464no—
221C23×C4—732no—
222C23×C4—732no—
223C23×C4—732no—
224C23×C4—732no—
225C23×C4C22×D7311yescharacteristic
226C23×C4C22×D7311yescharacteristic
227C23×C4C22×D7314yes—
228C23×C4C22×D7311yescharacteristic
229#125—14632no—
230C4×Q8—14632no—
231C2×C4⋊C4—2928no—
232C2×C4⋊C4—7316no—
233C2×C4⋊C4—738no—
234C2×C4⋊C4—734no—
235C2×C4⋊C4—738no—
236C2×C4⋊C4—738no—
237C2×C4⋊C4—734no—
238C2×C4⋊C4—734no—
239C2×C4⋊C4—7316no—
240C2×C4⋊C4—14616no—
241C2×C4⋊C4—1468no—
242C2×C4⋊C4—1464no—
243C2×C4⋊C4—1468no—
244C2×C4⋊C4—1468no—
245C2×C4⋊C4—1464no—
246C2×C4⋊C4—1464no—
247C2×C4⋊C4Dic146116yescomplemented
248C2×C4⋊C4C22×D7318yes—
249C2×C42—44no—
250C2×C42—738no—
251C2×C42—1468no—
252C2×C42D29218yes—
253C2×C42C22×D7318yes—
254C2×C42C22×D7314yes—
255#95—7316no—
256#95—7316no—
257#95C22×D7318yes—
258C24C23×D7311yescharacteristic
259C2×Q8—14616no—
260C2×Q8—14632no—
261C2×Q8—1468no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—738no—
268C22×C4—734no—
269C22×C4—734no—
270C22×C4—734no—
271C22×C4—734no—
272C22×C4—1464no—
273C22×C4—1464no—
274C22×C4—1464no—
275C22×C4—1468no—
276C22×C4—1464no—
277C22×C4—1464no—
278C22×C4—1464no—
279C22×C4—1464no—
280C22×C4—1464no—
281C22×C4—1464no—
282C22×C4C2×D29214yes—
283C22×C4C2×D29214yes—
284C22×C4C2×D29214yes—
285C22×C4C2×Dic146116yes—
286C22×C4C2×Dic14618yes—
287C22×C4C23×D7312yes—
288C22×C4C23×D7312yes—
289C22×C4C23×D7318yes—
290C22×C4C23×D7312yes—
291C4⋊C4—29216no—
292C4⋊C4—14632no—
293C4⋊C4—14616no—
294C4⋊C4—14616no—
295C4⋊C4—14616no—
296C4⋊C4—1468no—
297C4⋊C4—1468no—
298C4⋊C4—1468no—
299C4⋊C4C2×Dic146132yescomplemented
300C42—48no—
301C42—14616no—
302C42C2×D292116yes—
303C23C22×D29212yes—
304C23C22×Dic14614yes—
305C23#41×D7314yes—
306C23#1455663612yes—
307C23#1455663912yes—
308C23C24×D7311yescharacteristic
309Q8—14632no—
310C2×C4—28no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—28no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—232no—
317C2×C4—28no—
318C2×C4—2928no—
319C2×C4—2928no—
320C2×C4—2928no—
321C2×C4—738no—
322C2×C4—1464no—
323C2×C4—1464no—
324C2×C4—1464no—
325C2×C4—1468no—
326C2×C4—1464no—
327C2×C4—1464no—
328C2×C4—1464no—
329C2×C4—1464no—
330C2×C4—1464no—
331C2×C4—1464no—
332C2×C4—14616no—
333C2×C4#14556591116yes—
334C2×C4C22×D29214yes—
335C2×C4C22×D29214yes—
336C2×C4C22×D29214yes—
337C2×C4C22×Dic146116yes—
338C2×C4C22×Dic14618yes—
339C22#1798909914yes—
340C22C2×#1455659114yes—
341C22#1798916814yes—
342C22#1798929618yes—
343C22#1798931318yes—
344C22C23×D29211yescharacteristic
345C22C23×Dic14612yes—
346C22C2×#41×D7312yes—
347C22C2×#1455663611yescharacteristic
348C22C2×#1455663911yescharacteristic
349C4—24no—
350C4—24no—
351C4—216no—
352C4—2924no—
353C4—2924no—
354C4—2924no—
355C4—1468no—
356C4C2×#1455659118yes—
357C2#2261531418yesdirect product factor
358C2C2×#1798909911yescharacteristic
359C2C22×#1455659111yescharacteristic
360C2C2×#1798916811yescharacteristic
361C2C2×#1798929612yes—
362C2C2×#1798931312yes—
subgroupG/H
centerC24C23×D73
commutator, cosocleC23×C73C25
FrattiniC23C24×D73
FittingC2×C4×C4⋊C4×C73C2
socle, prime radicalC24×C73C24
hypercenterC2×C4×C4⋊C4D73
hypocenterC73C2×#2584
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22615314C2direct (N × H)
69. C2×C4⋊C4×C73Q8semidirect
118. C4⋊C4×C73C2×Q8semidirect
176. C2×C73#2584semidirect
191. C73C2×#2584semidirect
247. C2×C4⋊C4Dic146semidirect
299. C4⋊C4C2×Dic146semidirect
357. C2#22615314direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565832
C22×C4×Dic1462
#226153142
C2×#179885192
C2×#179885082
C2×#179884862
C2×#179884852
C2×C4×C4⋊C4×C732
C2×#258473