Finite Group Databasebeta

← C2×#22561351 #30883114

Subgroups — C2×#22561351 #30883114

Order 17152.

1518 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524043C211yesmaximal, characteristic
2C22×#14523968C212yesmaximal
3C2×#16863302C211yesmaximal, characteristic
4#22561351C218yesmaximal, direct product factor
5C2×#16863159C212yesmaximal
6C2×#336×C67C211yesmaximal, characteristic
7C23×Dic134—21no—
8C23×Dic134C2211yescharacteristic
9C2×#14524043—28no—
10C2×#14524043—22no—
11C2×#14524043C2214yes—
12C22×C4⋊Dic67—21no—
13C22×C4⋊Dic67C2211yescharacteristic
14C22×C4⋊Dic67C2212yes—
15C2×#14523968—28no—
16C2×#14523968C2218yes—
17#16863302—22no—
18#16863302C2214yes—
19C2×C67⋊#95—21no—
20C2×C67⋊#95—21no—
21#16863159C2218yes—
22C23×C8×C67C2212yes—
23C22×C42×C67C2211yescharacteristic
24#336×C67C2214yes—
25C22×Dic134—28no—
26C22×Dic134—21no—
27C22×Dic134—24no—
28C22×Dic134—24no—
29C22×Dic134—22no—
30C22×Dic134—24no—
31C22×Dic134—44no—
32C22×Dic134—44no—
33C22×Dic134Q814yescomplemented
34C22×Dic134C2312yes—
35C23×Dic67—21no—
36C23×Dic67—22no—
37C23×Dic67—21no—
38#14524043—216no—
39#14524043—216no—
40C2×C4⋊Dic67—28no—
41C2×C4⋊Dic67—24no—
42C2×C4⋊Dic67—24no—
43C2×C4⋊Dic67—22no—
44C2×C4⋊Dic67—21no—
45C2×C4⋊Dic67—24no—
46C2×C4⋊Dic67D418yes—
47C2×C4⋊Dic67C2312yes—
48C2×C4⋊Dic67C2314yes—
49#14523968—216no—
50C67⋊#95—24no—
51C67⋊#95—24no—
52C23×C4×C67D411yescharacteristic
53C23×C4×C67D411yescharacteristic
54C23×C4×C67C2311yescharacteristic
55C22×C8×C67—24no—
56C22×C8×C67—24no—
57C22×C8×C67—24no—
58C22×C8×C67C2314yes—
59C2×C42×C67—24no—
60C2×C42×C67D412yes—
61C2×C42×C67C2312yes—
62C2×Dic134—216no—
63C2×Dic134—24no—
64C2×Dic134—24no—
65C2×Dic134—216no—
66C2×Dic134—24no—
67C2×Dic134—22no—
68C2×Dic134—48no—
69C2×Dic134—432no—
70C2×Dic134—42no—
71C2×Dic134—48no—
72C2×Dic134—42no—
73C2×Dic134—48no—
74C2×Dic134C2×Q814yescomplemented
75C22×Dic67—28no—
76C22×Dic67—24no—
77C22×Dic67—22no—
78C22×Dic67—24no—
79C22×Dic67—22no—
80C22×Dic67—42no—
81C22×Dic67—42no—
82C22×Dic67—44no—
83C22×Dic67—44no—
84C22×Dic67—42no—
85C22×Dic67—42no—
86C22×Dic67—42no—
87C4⋊Dic67—216no—
88C4⋊Dic67—28no—
89C4⋊Dic67—24no—
90C4⋊Dic67—24no—
91C4⋊Dic67—42no—
92C4⋊Dic67C2×D418yes—
93C24×C67C2×D411yescharacteristic
94C22×C4×C67—22no—
95C22×C4×C67—22no—
96C22×C4×C67—22no—
97C22×C4×C67—22no—
98C22×C4×C67SD1614yes—
99C22×C4×C67Q1614yes—
100C22×C4×C67C2×D414yes—
101C22×C4×C67C2×D412yes—
102C22×C4×C67C2×D412yes—
103C22×C4×C67C2×D411yescharacteristic
104C22×C4×C67C2×Q812yes—
105C22×C4×C67#4112yes—
106C22×C4×C67#4114yes—
107C22×C4×C67C2411yescharacteristic
108C2×C8×C67—24no—
109C2×C8×C67—24no—
110C2×C8×C67—24no—
111C2×C8×C67—48no—
112C42×C67—24no—
113C42×C67—24no—
114C23×C67#12012yes—
115C23×C67C22⋊Q812yes—
116C23×C67#12412yes—
117C23×C67C2×SD1612yes—
118C23×C67C2×Q1612yes—
119C23×C67AGL(1,ℤ/8)12yes—
120C23×C67#13712yes—
121C23×C67C22×D411yescharacteristic
122C2×C4×C67—22no—
123C2×C4×C67—22no—
124C2×C4×C67—22no—
125C2×C4×C67—22no—
126C2×C4×C67—28no—
127C2×C4×C67—28no—
128C2×C4×C67#12012yes—
129C2×C4×C67C22⋊Q818yes—
130C2×C4×C67C22⋊Q812yes—
131C2×C4×C67#12418yes—
132C2×C4×C67#12412yes—
133C2×C4×C67C2×SD1614yes—
134C2×C4×C67C2×Q1614yes—
135C2×C4×C67C22×D412yes—
136C2×C4×C67C22×Q811yescharacteristic
137C2×C4×C67C2×#4111yescharacteristic
138C2×C4×C67C2×#4112yes—
139C2×Dic67—28no—
140C2×Dic67—24no—
141C2×Dic67—42no—
142C2×Dic67—42no—
143C2×Dic67—44no—
144C2×Dic67—44no—
145C2×Dic67—48no—
146C2×Dic67—416no—
147C2×Dic67—42no—
148C2×Dic67—42no—
149C2×Dic67—42no—
150C2×Dic67—84no—
151Dic134—28no—
152Dic134—432no—
153Dic134—44no—
154Dic134—48no—
155C8×C67—44no—
156C22×C67#39314yes—
157C22×C67#45014yes—
158C22×C67#45114yes—
159C22×C67#47514yes—
160C22×C67#47714yes—
161C22×C67#48714yes—
162C22×C67#48914yes—
163C22×C67C2×#12011yescharacteristic
164C22×C67C2×C22⋊Q811yescharacteristic
165C22×C67C2×#12411yescharacteristic
166C22×C67C22×SD1611yescharacteristic
167C22×C67C22×Q1611yescharacteristic
168C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
169C22×C67C2×#13711yescharacteristic
170C4×C67—24no—
171C4×C67—24no—
172C4×C67#39314yes—
173C4×C67C2×C22⋊Q812yes—
174C4×C67C2×#12412yes—
175Dic67—44no—
176Dic67—48no—
177Dic67—82no—
178C2×#2024—671nomaximal, Sylow
179C2×C67#202418yescomplemented
180C2×C67C2×#39311yescharacteristic
181C2×C67C2×#45011yescharacteristic
182C2×C67C2×#45111yescharacteristic
183C2×C67C2×#47511yescharacteristic
184C2×C67C2×#47711yescharacteristic
185C2×C67C2×#48711yescharacteristic
186C2×C67C2×#48911yescharacteristic
187C22×C4⋊Q8—671no—
188C22×#103—672no—
189C2×#391—671no—
190#2024—678no—
191C2×#340—672no—
192C2×#336D6711yescharacteristic
193C67C2×#202411yescharacteristic, complemented, Sylow
194C23×Q8—671no—
195C23×Q8—1341no—
196C23×C8C2×D6712yes—
197C2×C4⋊Q8—674no—
198C2×C4⋊Q8—1348no—
199C2×C4⋊Q8—1342no—
200C22×C4⋊C4—672no—
201C22×C4⋊C4—671no—
202C22×C4⋊C4—1341no—
203C22×C42C2×D6711yescharacteristic
204C2×#103—678no—
205C2×#103—1348no—
206#391—674no—
207#391—1342no—
208C2×#95—1341no—
209C2×#95—1341no—
210#340—678no—
211#336C2×D6714yes—
212C22×Q8—2684no—
213C22×Q8—2684no—
214C22×Q8—674no—
215C22×Q8—672no—
216C22×Q8—1348no—
217C22×Q8—1344no—
218C22×Q8—1344no—
219C22×Q8—1344no—
220C22×Q8—1341no—
221C22×Q8—1342no—
222C23×C4—1341no—
223C23×C4—1342no—
224C23×C4—1341no—
225C23×C4D26811yescharacteristic
226C23×C4D26811yescharacteristic
227C23×C4C22×D6711yescharacteristic
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C22×D6714yes—
232C4⋊Q8—13416no—
233C4⋊Q8—13416no—
234C2×C4⋊C4—678no—
235C2×C4⋊C4—674no—
236C2×C4⋊C4—672no—
237C2×C4⋊C4—1341no—
238C2×C4⋊C4—1344no—
239C2×C4⋊C4—1348no—
240C2×C4⋊C4—1344no—
241C2×C4⋊C4—1342no—
242C2×C4⋊C4—1344no—
243C2×C42—24no—
244C2×C42D26812yes—
245C2×C42C22×D6712yes—
246#103—13416no—
247#95—1344no—
248#95—1344no—
249C24C2×D26811yescharacteristic
250C2×Q8—26832no—
251C2×Q8—2688no—
252C2×Q8—2682no—
253C2×Q8—2688no—
254C2×Q8—2682no—
255C2×Q8—2688no—
256C2×Q8—674no—
257C2×Q8—13416no—
258C2×Q8—13416no—
259C2×Q8—1344no—
260C2×Q8—1344no—
261C2×Q8—1344no—
262C2×Q8—1342no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—2682no—
268C22×C4—2682no—
269C22×C4—2684no—
270C22×C4—2684no—
271C22×C4—2682no—
272C22×C4—2682no—
273C22×C4—2682no—
274C22×C4—1342no—
275C22×C4—1348no—
276C22×C4—1344no—
277C22×C4—1342no—
278C22×C4—1344no—
279C22×C4#1176161714yes—
280C22×C4Dic26814yes—
281C22×C4D4×D6714yes—
282C22×C4C2×D26812yes—
283C22×C4C2×D26812yes—
284C22×C4C2×D26811yescharacteristic
285C22×C4Q8×D6712yes—
286C22×C4#1176162712yes—
287C22×C4#1176162814yes—
288C22×C4C23×D6711yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—48no—
293C4⋊C4—2682no—
294C4⋊C4—678no—
295C4⋊C4—13416no—
296C4⋊C4—1344no—
297C4⋊C4—1348no—
298C4⋊C4—1344no—
299C42—24no—
300C42—24no—
301C23#1452401312yes—
302C23#1452402312yes—
303C23#1452403112yes—
304C23C2×#1176161712yes—
305C23C2×Dic26812yes—
306C23#1452407112yes—
307C23#1452407612yes—
308C23C22×D26811yescharacteristic
309Q8—26832no—
310Q8—2688no—
311Q8—2684no—
312Q8—1348no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—26816no—
320C2×C4—2688no—
321C2×C4—2682no—
322C2×C4—2682no—
323C2×C4—2684no—
324C2×C4—2684no—
325C2×C4—2682no—
326C2×C4—2682no—
327C2×C4—2682no—
328C2×C4—5364no—
329C2×C4—1348no—
330C2×C4—1344no—
331C2×C4#1452401312yes—
332C2×C4#1452402118yes—
333C2×C4#1452402312yes—
334C2×C4#1452402918yes—
335C2×C4#1452403112yes—
336C2×C4C2×#1176161714yes—
337C2×C4C2×Dic26814yes—
338C2×C4C2×D4×D6712yes—
339C2×C4C2×Q8×D6711yescharacteristic
340C2×C4C2×#1176162711yescharacteristic
341C2×C4C2×#1176162812yes—
342C8—44no—
343C22#1686330914yes—
344C22#1686354914yes—
345C22#1686355414yes—
346C22#1686368214yes—
347C22#1686369214yes—
348C22#1686374714yes—
349C22#1686375614yes—
350C22C2×#1452401311yescharacteristic
351C22C2×#1452402311yescharacteristic
352C22C2×#1452403111yescharacteristic
353C22C22×#1176161711yescharacteristic
354C22C22×Dic26811yescharacteristic
355C22C2×#1452407111yescharacteristic
356C22C2×#1452407611yescharacteristic
357C4—24no—
358C4—24no—
359C4—2684no—
360C4—2688no—
361C4—5362no—
362C4#1686330914yes—
363C4C2×#1452402112yes—
364C4C2×#1452402912yes—
365C2#2256135118yesdirect product factor
366C2C2×#1686330911yescharacteristic
367C2C2×#1686354911yescharacteristic
368C2C2×#1686355411yescharacteristic
369C2C2×#1686368211yescharacteristic
370C2C2×#1686369211yescharacteristic
371C2C2×#1686374711yescharacteristic
372C2C2×#1686375611yescharacteristic
subgroupG/H
centerC24C2×D268
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC2×#336×C67C2
socle, prime radicalC24×C67C2×D4
hypercenterC2×#336D67
hypocenterC67C2×#2024
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
4. #22561351C2direct (N × H)
33. C22×Dic134Q8semidirect
74. C2×Dic134C2×Q8semidirect
179. C2×C67#2024semidirect
193. C67C2×#2024semidirect
365. C2#22561351direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240432
C22×#145239682
C2×#168633022
#225613512
C2×#168631592
C2×#336×C672
C2×#202467