Finite Group Databasebeta

← C2×#22615297 #33257312

Subgroups — C2×#22615297 #33257312

Order 18688.

2614 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×C4×Dic146C211yesmaximal, characteristic
2#22615297C2116yesmaximal, direct product factor
3C2×#17988525C212yesmaximal
4C2×#17988506C212yesmaximal
5C2×#17988493C212yesmaximal
6C2×#17988485C212yesmaximal
7C2×#17988468C214yesmaximal
8C2×#17988453C211yesmaximal, characteristic
9C22×C4×Q8×C73C211yesmaximal, characteristic
10C23×Dic146C2211yescharacteristic
11C2×C4×Dic146—28no—
12C2×C4×Dic146C2218yes—
13C22×C4⋊Dic73C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924007C2214yes—
16C22×#11924007C2212yes—
17C22×C4×Dic73C2212yes—
18#17988525C22116yes—
19#17988506C22116yes—
20#17988493C22116yes—
21#17988485C22116yes—
22#17988468C22132yes—
23#17988453C2218yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2214yes—
27C2×#14556498C2214yes—
28C2×#14556498C2214yes—
29C23×Q8×C73C2211yescharacteristic
30C2×C4×Q8×C73—28no—
31C2×C4×Q8×C73C2218yes—
32C22×C4⋊C4×C73C2212yes—
33C22×C4⋊C4×C73C2211yescharacteristic
34C22×C42×C73C2212yes—
35C22×C42×C73C2211yescharacteristic
36C22×Dic146—24no—
37C22×Dic146—24no—
38C22×Dic146Q818yescomplemented
39C22×Dic146C2314yes—
40C23×Dic73C2314yes—
41C23×Dic73C2312yes—
42C23×Dic73C2312yes—
43C4×Dic146—232no—
44C2×C4⋊Dic73—24no—
45C2×C4⋊Dic73—28no—
46C2×C4⋊Dic73C2318yes—
47C2×C4⋊Dic73C2314yes—
48C2×#11924007—28no—
49C2×#11924007—216no—
50C2×#11924007C23116yes—
51C2×#11924007C2318yes—
52C2×C4×Dic73—28no—
53C2×C4×Dic73C2318yes—
54#14556498C2318yes—
55#14556498C2318yes—
56#14556498C23116yes—
57#14556498C23116yes—
58#14556498C23116yes—
59C22×Q8×C73—24no—
60C22×Q8×C73—28no—
61C22×Q8×C73C2314yes—
62C23×C4×C73C2312yes—
63C23×C4×C73C2311yescharacteristic
64C23×C4×C73C2311yescharacteristic
65C23×C4×C73C2312yes—
66C23×C4×C73C2311yescharacteristic
67C4×Q8×C73—232no—
68C2×C4⋊C4×C73—28no—
69C2×C4⋊C4×C73—24no—
70C2×C4⋊C4×C73C2318yes—
71C2×C4⋊C4×C73C2314yes—
72C2×C42×C73—28no—
73C2×C42×C73D418yes—
74C2×C42×C73C2318yes—
75C2×C42×C73C2314yes—
76C2×Dic146—232no—
77C2×Dic146—28no—
78C2×Dic146—28no—
79C2×Dic146C2×Q8116yescomplemented
80C22×Dic73—24no—
81C22×Dic73—24no—
82C22×Dic73—24no—
83C22×Dic73—28no—
84C22×Dic73—28no—
85C22×Dic73—28no—
86C22×Dic73—28no—
87C22×Dic73C2×Q818yes—
88C22×Dic73C2418yes—
89C22×Dic73C2414yes—
90C22×Dic73C2414yes—
91C4⋊Dic73—28no—
92C4⋊Dic73—216no—
93#11924007—216no—
94#11924007—232no—
95C4×Dic73—216no—
96C24×C73C2411yescharacteristic
97C2×Q8×C73—232no—
98C2×Q8×C73—28no—
99C2×Q8×C73—216no—
100C22×C4×C73—24no—
101C22×C4×C73—24no—
102C22×C4×C73—24no—
103C22×C4×C73—24no—
104C22×C4×C73—24no—
105C22×C4×C73—28no—
106C22×C4×C73C2×D414yes—
107C22×C4×C73C2×D414yes—
108C22×C4×C73C2×D414yes—
109C22×C4×C73C2×Q814yes—
110C22×C4×C73#4118yes—
111C22×C4×C73#4114yes—
112C22×C4×C73C2414yes—
113C22×C4×C73C2412yes—
114C22×C4×C73C2412yes—
115C22×C4×C73C2414yes—
116C22×C4×C73C2412yes—
117C4⋊C4×C73—216no—
118C4⋊C4×C73—28no—
119C42×C73—216no—
120C42×C73C2×D4116yes—
121C23×C73C22×D412yes—
122C23×C73C22×Q812yes—
123C23×C73C2×#4114yes—
124C23×C73C2×#4112yes—
125C23×C732−1+414yes—
126C23×C73C2511yescharacteristic
127Q8×C73—232no—
128C2×C4×C73—28no—
129C2×C4×C73—216no—
130C2×C4×C73—28no—
131C2×C4×C73—24no—
132C2×C4×C73—24no—
133C2×C4×C73—24no—
134C2×C4×C73—24no—
135C2×C4×C73—24no—
136C2×C4×C73—28no—
137C2×C4×C73—48no—
138C2×C4×C73C22⋊Q8116yes—
139C2×C4×C73C22×D414yes—
140C2×C4×C73C22×D414yes—
141C2×C4×C73C22×D414yes—
142C2×C4×C73C22×Q814yes—
143C2×C4×C73C2×#4118yes—
144C2×C4×C73C2×#4114yes—
145C2×Dic73—216no—
146C2×Dic73—24no—
147C2×Dic73—24no—
148C2×Dic73—24no—
149C2×Dic73—28no—
150C2×Dic73—28no—
151C2×Dic73—28no—
152C2×Dic73—28no—
153C2×Dic73—48no—
154C2×Dic73—416no—
155C2×Dic73C22×Q818yes—
156Dic146—232no—
157C22×C73C2×C22⋊Q814yes—
158C22×C73#52514yes—
159C22×C73#53614yes—
160C22×C73#55618yes—
161C22×C73#55718yes—
162C22×C73C23×D411yescharacteristic
163C22×C73C23×Q811yescharacteristic
164C22×C73C22×#4112yes—
165C22×C73C22×#4111yescharacteristic
166C22×C73C2×2−1+412yes—
167C4×C73—24no—
168C4×C73—28no—
169C4×C73—24no—
170C4×C73—44no—
171C4×C73C2×C22⋊Q818yes—
172Dic73—28no—
173Dic73—44no—
174Dic73—48no—
175C2×#2581—731nomaximal, Sylow
176C2×C73#258118yescomplemented
177C2×C73C22×C22⋊Q811yescharacteristic
178C2×C73C2×#52511yescharacteristic
179C2×C73C2×#53611yescharacteristic
180C2×C73C2×#55612yes—
181C2×C73C2×#55712yes—
182C22×C4×Q8—731no—
183C22×C4×Q8D7311yescharacteristic
184#2581—7316no—
185C2×#400—732no—
186C2×#395—732no—
187C2×#391—732no—
188C2×#389—732no—
189C2×#387—734no—
190C2×#382—731no—
191C73C2×#258111yescharacteristic, complemented, Sylow
192C23×Q8—731no—
193C23×Q8C2×D7311yescharacteristic
194C2×C4×Q8—28no—
195C2×C4×Q8—738no—
196C2×C4×Q8—1468no—
197C2×C4×Q8C2×D7318yes—
198C22×C4⋊C4—732no—
199C22×C4⋊C4—734no—
200C22×C4⋊C4—731no—
201C22×C4⋊C4—732no—
202C22×C4⋊C4C2×D7311yescharacteristic
203C22×C4⋊C4C2×D7312yes—
204C22×C42—732no—
205C22×C42C2×D7312yes—
206C22×C42C2×D7311yescharacteristic
207#400—7316no—
208#395—7316no—
209#391—7316no—
210#389—7316no—
211#387—7332no—
212#382—738no—
213C2×#95—732no—
214C2×#95—734no—
215C2×#95—734no—
216C2×#95—734no—
217C2×#95—732no—
218C22×Q8—24no—
219C22×Q8—28no—
220C22×Q8—738no—
221C22×Q8—734no—
222C22×Q8—1464no—
223C22×Q8—1464no—
224C22×Q8C22×D7314yes—
225C23×C4—732no—
226C23×C4—732no—
227C23×C4—734no—
228C23×C4C22×D7312yes—
229C23×C4C22×D7311yescharacteristic
230C23×C4C22×D7311yescharacteristic
231C23×C4C22×D7312yes—
232C23×C4C22×D7311yescharacteristic
233C4×Q8—232no—
234C4×Q8—14632no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—738no—
238C2×C4⋊C4—734no—
239C2×C4⋊C4—738no—
240C2×C4⋊C4—7316no—
241C2×C4⋊C4—1468no—
242C2×C4⋊C4—1464no—
243C2×C4⋊C4—14616no—
244C2×C4⋊C4—1468no—
245C2×C4⋊C4C22×D7318yes—
246C2×C4⋊C4C22×D7314yes—
247C2×C42—28no—
248C2×C42—738no—
249C2×C42—1468no—
250C2×C42C22⋊D7318yes—
251C2×C42C22×D7318yes—
252C2×C42C22×D7314yes—
253#95—738no—
254#95—738no—
255#95—7316no—
256#95—7316no—
257#95—7316no—
258C24C23×D7311yescharacteristic
259C2×Q8—232no—
260C2×Q8—28no—
261C2×Q8—216no—
262C2×Q8—7316no—
263C2×Q8—14632no—
264C2×Q8—1468no—
265C2×Q8—1468no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—738no—
273C22×C4—734no—
274C22×C4—738no—
275C22×C4—734no—
276C22×C4—1464no—
277C22×C4—1468no—
278C22×C4—1468no—
279C22×C4—1468no—
280C22×C4—1468no—
281C22×C4—1464no—
282C22×C4—1464no—
283C22×C4C2×C22⋊D7314yes—
284C22×C4C2×C22⋊D7314yes—
285C22×C4C2×C22⋊D7314yes—
286C22×C4Q8×D7314yes—
287C22×C4#1192402818yes—
288C22×C4#1192402914yes—
289C22×C4C23×D7314yes—
290C22×C4C23×D7312yes—
291C22×C4C23×D7312yes—
292C22×C4C23×D7314yes—
293C22×C4C23×D7312yes—
294C4⋊C4—216no—
295C4⋊C4—28no—
296C4⋊C4—14616no—
297C4⋊C4—1468no—
298C4⋊C4—14632no—
299C4⋊C4—14616no—
300C42—216no—
301C42—14616no—
302C42C2×C22⋊D73116yes—
303C23C22×C22⋊D7312yes—
304C23C2×Q8×D7312yes—
305C23C2×#1192402814yes—
306C23C2×#1192402912yes—
307C23#1455663814yes—
308C23C24×D7311yescharacteristic
309Q8—232no—
310Q8—14632no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—216no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—48no—
321C2×C4—29216no—
322C2×C4—2928no—
323C2×C4—738no—
324C2×C4—1464no—
325C2×C4—1468no—
326C2×C4—1468no—
327C2×C4—1468no—
328C2×C4—1468no—
329C2×C4—1464no—
330C2×C4—1464no—
331C2×C4—14616no—
332C2×C4#14556567116yes—
333C2×C4C22×C22⋊D7314yes—
334C2×C4C22×C22⋊D7314yes—
335C2×C4C22×C22⋊D7314yes—
336C2×C4C2×Q8×D7314yes—
337C2×C4C2×#1192402818yes—
338C2×C4C2×#1192402914yes—
339C22C2×#1455656714yes—
340C22#1798909814yes—
341C22#1798915814yes—
342C22#1798931018yes—
343C22#1798931518yes—
344C22C23×C22⋊D7311yescharacteristic
345C22C22×Q8×D7311yescharacteristic
346C22C22×#1192402812yes—
347C22C22×#1192402911yescharacteristic
348C22C2×#1455663812yes—
349C4—24no—
350C4—24no—
351C4—28no—
352C4—44no—
353C4—2924no—
354C4—2928no—
355C4—1468no—
356C4C2×#1455656718yes—
357C2#2261529718yesdirect product factor
358C2C22×#1455656711yescharacteristic
359C2C2×#1798909811yescharacteristic
360C2C2×#1798915811yescharacteristic
361C2C2×#1798931012yes—
362C2C2×#1798931512yes—
subgroupG/H
centerC24C23×D73
commutator, cosocleC23×C73C25
FrattiniC23C24×D73
FittingC22×C4×Q8×C73C2
socle, prime radicalC24×C73C24
hypercenterC22×C4×Q8D73
hypocenterC73C2×#2581
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22615297C2direct (N × H)
38. C22×Dic146Q8semidirect
79. C2×Dic146C2×Q8semidirect
176. C2×C73#2581semidirect
191. C73C2×#2581semidirect
357. C2#22615297direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Dic1462
#226152972
C2×#179885252
C2×#179885062
C2×#179884932
C2×#179884852
C2×#179884682
C2×#179884532
C22×C4×Q8×C732
C2×#258173