Finite Group Databasebeta

← C2×#22609356 #33068271

Subgroups — C2×#22609356 #33068271

Order 18688.

758 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988939C211yesmaximal, characteristic, complemented
2C2×#17988867C211yesmaximal, characteristic
3C2×D4×#10596831C211yesmaximal, characteristic
4#22609356C218yesmaximal, direct product factor
5C2×#17988337C211yesmaximal, characteristic, complemented
6C2×#17988333C211yesmaximal, characteristic, complemented
7C2×#17988332C211yesmaximal, characteristic, complemented
8C2×#461×C73C211yesmaximal, characteristic
9C23×#10596831—21no—
10C2×#14556591C2211yescharacteristic
11#17988939—22no—
12#17988939C2214yescomplemented
13#17988867—22no—
14#17988867C2214yes—
15D4×#10596831—22no—
16D4×#10596831C2214yes—
17C2×#14556524—21no—
18C2×C4⋊#10596831C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556514—21no—
21C2×#14556514—21no—
22C2×#14556512—21no—
23C2×C22⋊#10596831—21no—
24C2×C4×#10596831C2211yescharacteristic
25#17988337C2214yescomplemented
26#17988333C2214yescomplemented
27#17988332C2214yescomplemented
28C22×SD16×C73—21no—
29C2×C4⋊Q8×C73C2211yescharacteristic
30C2×C4×D4×C73C2211yescharacteristic
31#461×C73—22no—
32#461×C73C2214yes—
33C2×C4⋊C8×C73C2211yescharacteristic
34C2×#103×C73—21no—
35C22×Dic146—21no—
36C22×#10596831—24no—
37C22×#10596831—21no—
38C22×#10596831—21no—
39C22×#10596831—44no—
40C22×#10596831D411yescharacteristic
41C22×#10596831D411yescharacteristic
42#14556591—21no—
43#14556591—21no—
44#14556591—21no—
45#14556591C2312yes—
46C2×C4⋊Dic73—21no—
47C2×C4⋊Dic73—21no—
48#14556524—24no—
49C4⋊#10596831—21no—
50C4⋊#10596831—21no—
51C4⋊#10596831C2312yes—
52C4⋊#10596831C2312yes—
53#14556514—24no—
54#14556514—24no—
55#14556512—24no—
56C22⋊#10596831—24no—
57C4×#10596831D412yes—
58C4×#10596831C2312yes—
59C22×Q8×C73—21no—
60C22×D4×C73D411yescharacteristic
61C23×C4×C73—21no—
62C2×SD16×C73—24no—
63C2×SD16×C73—44no—
64C22×C8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73C2312yes—
69C4×D4×C73—21no—
70C4×D4×C73—21no—
71C4×D4×C73D412yes—
72C4×D4×C73C2312yes—
73C2×C4⋊C4×C73—21no—
74C2×C4⋊C4×C73—21no—
75C2×C4⋊C4×C73D411yescharacteristic
76C2×C22⋊C4×C73—21no—
77C2×C42×C73C2311yescharacteristic
78C4⋊C8×C73—21no—
79C4⋊C8×C73C2312yes—
80#103×C73—24no—
81C2×Dic146—22no—
82C2×Dic146—42no—
83C2×Dic146—42no—
84C2×Dic146—41no—
85C22×Dic73—41no—
86C2×#10596831—22no—
87C2×#10596831—22no—
88C2×#10596831—22no—
89C2×#10596831—22no—
90C2×#10596831—48no—
91C2×#10596831—42no—
92C2×#10596831—42no—
93C2×#10596831C2×D412yes—
94C2×#10596831C2×D412yes—
95C4⋊Dic73—22no—
96C4⋊Dic73—22no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99C24×C73—21no—
100C2×Q8×C73—22no—
101C2×Q8×C73—41no—
102C2×Q8×C73—42no—
103C2×Q8×C73—42no—
104C2×D4×C73—21no—
105C2×D4×C73—22no—
106C2×D4×C73SD1614yes—
107C2×D4×C73C2×D412yes—
108C22×C4×C73—21no—
109C22×C4×C73—22no—
110C22×C4×C73—22no—
111C22×C4×C73—42no—
112C22×C4×C73—42no—
113C22×C4×C73—41no—
114C22×C4×C73C2×D411yescharacteristic
115C22×C4×C73C2×D411yescharacteristic
116C22×C4×C73#4111yescharacteristic
117SD16×C73—48no—
118C2×C8×C73—22no—
119C2×C8×C73—42no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—22no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73C2×D412yes—
126C22⋊C4×C73—22no—
127C22⋊C4×C73—22no—
128C42×C73C2×D411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73C2411yescharacteristic
132C23×C73—22no—
133C23×C73—24no—
134C23×C73—42no—
135C23×C73—42no—
136C23×C73#12111yescharacteristic
137Q8×C73—42no—
138Q8×C73—42no—
139Q8×C73—82no—
140D4×C73—22no—
141D4×C73—24no—
142D4×C73C2×SD1614yes—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—42no—
148C2×C4×C73—44no—
149C2×C4×C73—42no—
150C2×C4×C73—82no—
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12111yescharacteristic
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73C2×SD1612yes—
155C2×C4×C73#13512yes—
156C2×C4×C73#13512yes—
157C2×C4×C73AGL(1,ℤ/8)12yes—
158C2×C4×C73C22×D411yescharacteristic
159C2×C4×C73C22×D411yescharacteristic
160C2×C4×C73C2×#4111yescharacteristic
161C2×Dic73—42no—
162C2×Dic73—82no—
163Dic146—42no—
164Dic146—42no—
165Dic146—82no—
166C8×C73—42no—
167#10596831—22no—
168#10596831—22no—
169#10596831—42no—
170#10596831—42no—
171C22×C73—24no—
172C22×C73—48no—
173C22×C73—42no—
174C22×C73—42no—
175C22×C73#46312yes—
176C22×C73#46512yes—
177C22×C73#47312yes—
178C22×C73C2×#12111yescharacteristic
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—42no—
182C4×C73—82no—
183C4×C73#46312yes—
184C4×C73#46512yes—
185C4×C73C22×SD1611yescharacteristic
186C4×C73C2×#13511yescharacteristic
187C4×C73C2×#13511yescharacteristic
188C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
189Dic73—82no—
190C2×#1679—731nomaximal, Sylow
191C2×C73—44no—
192C2×C73#167914yescomplemented
193C2×C73C2×#46311yescharacteristic
194C2×C73C2×#46511yescharacteristic
195C2×C73C2×#47311yescharacteristic
196C2×#487—731no—
197C2×#474—731no—
198C2×#461D7311yescharacteristic
199C2×C8×D4—731no—
200#1679—738no—
201C2×#334—731no—
202C2×#332—731no—
203C2×#332—731no—
204C73C2×#167911yescharacteristic, complemented, Sylow
205C22×SD16—21no—
206C23×C8—1461no—
207C2×C4⋊Q8—731no—
208C2×C4⋊Q8C2×D7311yescharacteristic
209C2×C4×D4C2×D7311yescharacteristic
210#487—734no—
211#487—1462no—
212#474—734no—
213#474—1462no—
214#461—22no—
215#461C2×D7314yes—
216C8×D4—734no—
217C8×D4—1462no—
218C2×#107—1461no—
219C2×C4⋊C8—731no—
220C2×C4⋊C8—731no—
221C2×C4⋊C8C2×D7311yescharacteristic
222C2×#103—21no—
223C2×#103—1461no—
224C2×#103—1461no—
225C2×#102—1461no—
226C2×C22⋊C8—1461no—
227C2×C4×C8—731no—
228#334—734no—
229#332—734no—
230#332—734no—
231C22×Q8—21no—
232C22×Q8—1461no—
233C22×D4C22⋊D7311yescharacteristic
234C23×C4—21no—
235C2×SD16—24no—
236C2×SD16—44no—
237C22×C8—21no—
238C22×C8—2924no—
239C22×C8—731no—
240C22×C8—731no—
241C22×C8—1464no—
242C22×C8—1461no—
243C22×C8—1461no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—732no—
248C4⋊Q8—1461no—
249C4⋊Q8—1461no—
250C4⋊Q8—1461no—
251C4⋊Q8C22×D7312yes—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C22⋊D7312yes—
255C4×D4C22×D7312yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—1461no—
259C2×C4⋊C4—1461no—
260C2×C4⋊C4C22⋊D7311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C42C22×D7311yescharacteristic
263#107—1464no—
264C4⋊C8—21no—
265C4⋊C8—732no—
266C4⋊C8—732no—
267C4⋊C8—1461no—
268C4⋊C8—1461no—
269C4⋊C8C22×D7312yes—
270#103—24no—
271#103—1464no—
272#103—1464no—
273#102—1464no—
274C22⋊C8—1464no—
275C4×C8—732no—
276C4×C8—732no—
277C24—21no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—2921no—
283C2×Q8—2922no—
284C2×Q8—2922no—
285C2×Q8—1462no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4#1192401614yes—
289C2×D4C2×C22⋊D7312yes—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—2921no—
297C22×C4D4×D7311yescharacteristic
298C22×C4C2×C22⋊D7311yescharacteristic
299C22×C4#1192402711yescharacteristic
300SD16—48no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—2922no—
304C2×C8—2922no—
305C2×C8—2928no—
306C2×C8—732no—
307C2×C8—732no—
308C2×C8—1462no—
309C2×C8—1462no—
310C2×C8—1462no—
311C2×C8—1462no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—2921no—
318C4⋊C4—2921no—
319C4⋊C4—1462no—
320C4⋊C4—1462no—
321C4⋊C4C2×C22⋊D7312yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C42D4×D7311yescharacteristic
325C42C2×C22⋊D7311yescharacteristic
326C42#1192402711yescharacteristic
327C42C23×D7311yescharacteristic
328C23—24no—
329C23—22no—
330C23—42no—
331C23—42no—
332C23#1455656311yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—2922no—
337Q8—2922no—
338Q8—5842no—
339D4—24no—
340D4—22no—
341D4C2×#1192401614yes—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—2922no—
351C2×C4—5842no—
352C2×C4#1455656311yescharacteristic
353C2×C4#1455656311yescharacteristic
354C2×C4#1455656311yescharacteristic
355C2×C4C2×#1192401612yes—
356C2×C4#1455661112yes—
357C2×C4#1455661212yes—
358C2×C4#1455661812yes—
359C2×C4C2×D4×D7311yescharacteristic
360C2×C4C22×C22⋊D7311yescharacteristic
361C2×C4C2×#1192402711yescharacteristic
362C8—42no—
363C8—2922no—
364C8—2922no—
365C8—1462no—
366C8—1462no—
367C22—24no—
368C22—48no—
369C22—42no—
370C22—42no—
371C22#1798880412yes—
372C22#1798881412yes—
373C22#1798886012yes—
374C22C2×#1455656311yescharacteristic
375C4—21no—
376C4—21no—
377C4—42no—
378C4—82no—
379C4—5842no—
380C4#1798880412yes—
381C4#1798881412yes—
382C4C22×#1192401611yescharacteristic
383C4C2×#1455661111yescharacteristic
384C4C2×#1455661211yescharacteristic
385C4C2×#1455661811yescharacteristic
386C2—44no—
387C2#2260935614yesdirect product factor
388C2C2×#1798880411yescharacteristic
389C2C2×#1798881411yescharacteristic
390C2C2×#1798886011yescharacteristic
subgroupG/H
centerC23#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#461×C73C2
socleC23×C73#121
prime radicalC22×D4×C73D4
hypercenterC2×#461D73
hypocenterC73C2×#1679
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#17988939C2semidirect
4. #22609356C2direct (N × H)
5. C2×#17988337C2semidirect
6. C2×#17988333C2semidirect
7. C2×#17988332C2semidirect
12. #17988939C22semidirect
25. #17988337C22semidirect
26. #17988333C22semidirect
27. #17988332C22semidirect
192. C2×C73#1679semidirect
204. C73C2×#1679semidirect
387. C2#22609356direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179889392
C2×#179888672
C2×D4×#105968312
#226093562
C2×#179883372
C2×#179883332
C2×#179883322
C2×#461×C732
C2×#167973