Finite Group Databasebeta

← C2×#22609382 #33068285

Subgroups — C2×#22609382 #33068285

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988938C211yesmaximal, characteristic, complemented
2C2×#17988877C211yesmaximal, characteristic
3C2×#17988867C211yesmaximal, characteristic
4#22609382C218yesmaximal, direct product factor
5C2×#17988340C211yesmaximal, characteristic, complemented
6C2×#17988335C211yesmaximal, characteristic, complemented
7C2×#17988334C211yesmaximal, characteristic, complemented
8C2×C8×D4×C73C211yesmaximal, characteristic
9C2×#14556591C2211yescharacteristic
10C2×#14556591C2211yescharacteristic
11#17988938—22no—
12#17988938C2214yescomplemented
13#17988877—22no—
14#17988877C2214yes—
15#17988867—22no—
16#17988867C2214yes—
17C2×#14556524—21no—
18C2×#14556522—21no—
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556516—21no—
22C2×#14556516—21no—
23C2×#14556512—21no—
24C2×#14556512—21no—
25#17988340C2214yescomplemented
26#17988335C2214yescomplemented
27#17988334C2214yescomplemented
28C23×C8×C73—21no—
29C2×C4×D4×C73C2211yescharacteristic
30C8×D4×C73—22no—
31C8×D4×C73C2214yes—
32C2×C4⋊C8×C73C2211yescharacteristic
33C2×C22⋊C8×C73—21no—
34C2×C4×C8×C73C2211yescharacteristic
35C22×Dic146—21no—
36C22×Dic146—21no—
37C22×#10596831—21no—
38C22×#10596831—21no—
39#14556591—21no—
40#14556591—21no—
41#14556591—21no—
42#14556591—21no—
43#14556591—21no—
44#14556591—21no—
45#14556591C2312yes—
46#14556591C2312yes—
47C2×C4⋊Dic73—21no—
48C2×C4⋊Dic73—21no—
49C2×C4⋊Dic73—21no—
50C2×C4⋊Dic73—21no—
51#14556524—24no—
52#14556522—24no—
53C4⋊#10596831—21no—
54C4⋊#10596831—21no—
55C4⋊#10596831C2312yes—
56C4⋊#10596831C2312yes—
57#14556516—24no—
58#14556516—24no—
59#14556512—24no—
60#14556512—24no—
61C22×D4×C73D411yescharacteristic
62C23×C4×C73—21no—
63C22×C8×C73—21no—
64C22×C8×C73—24no—
65C22×C8×C73—44no—
66C22×C8×C73D411yescharacteristic
67C22×C8×C73D411yescharacteristic
68C4×D4×C73—21no—
69C4×D4×C73—21no—
70C4×D4×C73D412yes—
71C4×D4×C73C2312yes—
72C2×C4⋊C4×C73D411yescharacteristic
73C2×C22⋊C4×C73—21no—
74C2×C42×C73C2311yescharacteristic
75C4⋊C8×C73—21no—
76C4⋊C8×C73C2312yes—
77C22⋊C8×C73—24no—
78C4×C8×C73D412yes—
79C4×C8×C73C2312yes—
80C2×Dic146—22no—
81C2×Dic146—22no—
82C2×Dic146—42no—
83C2×Dic146—42no—
84C2×Dic146—41no—
85C2×Dic146—41no—
86C2×Dic146—82no—
87C22×Dic73—41no—
88C22×Dic73—41no—
89C2×#10596831—22no—
90C2×#10596831—22no—
91C2×#10596831—42no—
92C2×#10596831—42no—
93C4⋊Dic73—22no—
94C4⋊Dic73—22no—
95C4⋊Dic73—22no—
96C4⋊Dic73—22no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99C4⋊Dic73—41no—
100C4⋊Dic73—41no—
101C24×C73—21no—
102C2×D4×C73—21no—
103C2×D4×C73—22no—
104C2×D4×C73Q1614yes—
105C2×D4×C73C2×D412yes—
106C22×C4×C73—21no—
107C22×C4×C73—22no—
108C22×C4×C73—22no—
109C22×C4×C73—42no—
110C22×C4×C73—42no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4111yescharacteristic
114C2×C8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73—22no—
117C2×C8×C73—48no—
118C2×C8×C73—42no—
119C2×C8×C73C2×D412yes—
120C2×C8×C73C2×D412yes—
121C4⋊C4×C73—21no—
122C4⋊C4×C73C2×D412yes—
123C22⋊C4×C73—22no—
124C22⋊C4×C73—22no—
125C42×C73C2×D411yescharacteristic
126C42×C73C2×D411yescharacteristic
127C42×C73#4111yescharacteristic
128C42×C73C2411yescharacteristic
129C23×C73—22no—
130C23×C73—24no—
131C23×C73—42no—
132C23×C73—42no—
133C23×C73#12111yescharacteristic
134D4×C73—22no—
135D4×C73—24no—
136D4×C73C2×Q1614yes—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—42no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—44no—
143C2×C4×C73#12111yescharacteristic
144C2×C4×C73#12111yescharacteristic
145C2×C4×C73#12111yescharacteristic
146C2×C4×C73C2×Q1612yes—
147C2×C4×C73#13512yes—
148C2×C4×C73#13512yes—
149C2×C4×C73AGL(1,ℤ/8)12yes—
150C2×C4×C73C22×D411yescharacteristic
151C2×C4×C73C22×D411yescharacteristic
152C2×C4×C73C2×#4111yescharacteristic
153C2×Dic73—42no—
154C2×Dic73—42no—
155C2×Dic73—82no—
156C2×Dic73—82no—
157Dic146—42no—
158Dic146—42no—
159Dic146—82no—
160Dic146—82no—
161Dic146—82no—
162C8×C73—22no—
163C8×C73—22no—
164C8×C73—42no—
165#10596831—42no—
166#10596831—42no—
167C22×C73—24no—
168C22×C73—48no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73#46312yes—
172C22×C73#46712yes—
173C22×C73#47112yes—
174C22×C73C2×#12111yescharacteristic
175C4×C73—21no—
176C4×C73—21no—
177C4×C73—42no—
178C4×C73#46312yes—
179C4×C73#46712yes—
180C4×C73C22×Q1611yescharacteristic
181C4×C73C2×#13511yescharacteristic
182C4×C73C2×#13511yescharacteristic
183C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
184Dic73—82no—
185Dic73—82no—
186C2×#1683—731nomaximal, Sylow
187C2×C73—44no—
188C2×C73#168314yescomplemented
189C2×C73C2×#46311yescharacteristic
190C2×C73C2×#46711yescharacteristic
191C2×C73C2×#47111yescharacteristic
192C2×#487—731no—
193C2×#476—731no—
194C2×#474—731no—
195C2×C8×D4D7311yescharacteristic
196#1683—738no—
197C2×#335—731no—
198C2×#333—731no—
199C2×#332—731no—
200C73C2×#168311yescharacteristic, complemented, Sylow
201C23×C8—21no—
202C2×C4⋊Q8—731no—
203C2×C4⋊Q8—731no—
204C2×C4×D4C2×D7311yescharacteristic
205#487—734no—
206#487—1462no—
207#476—734no—
208#476—1462no—
209#474—734no—
210#474—1462no—
211C8×D4—22no—
212C8×D4C2×D7314yes—
213C2×#107—1461no—
214C2×#106—1461no—
215C2×C4⋊C8—731no—
216C2×C4⋊C8—731no—
217C2×C4⋊C8C2×D7311yescharacteristic
218C2×#103—1461no—
219C2×#103—1461no—
220C2×#102—1461no—
221C2×#102—1461no—
222C2×C22⋊C8—21no—
223C2×C4×C8C2×D7311yescharacteristic
224#335—734no—
225#333—734no—
226#332—734no—
227C22×Q8—1461no—
228C22×Q8—1461no—
229C22×D4D29211yescharacteristic
230C23×C4—21no—
231C22×C8—21no—
232C22×C8—24no—
233C22×C8—44no—
234C22×C8—1461no—
235C22×C8—1461no—
236C22×C8C22⋊D7311yescharacteristic
237C22×C8C22⋊D7311yescharacteristic
238C4⋊Q8—732no—
239C4⋊Q8—732no—
240C4⋊Q8—1461no—
241C4⋊Q8—1461no—
242C4⋊Q8—1461no—
243C4⋊Q8—1461no—
244C4⋊Q8—1461no—
245C4⋊Q8—1461no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4D29212yes—
249C4×D4C22×D7312yes—
250C2×C4⋊C4—1461no—
251C2×C4⋊C4—1461no—
252C2×C4⋊C4—1461no—
253C2×C4⋊C4—1461no—
254C2×C4⋊C4D29211yescharacteristic
255C2×C22⋊C4—21no—
256C2×C42C22×D7311yescharacteristic
257#107—1464no—
258#106—1464no—
259C4⋊C8—21no—
260C4⋊C8—732no—
261C4⋊C8—732no—
262C4⋊C8—1461no—
263C4⋊C8—1461no—
264C4⋊C8C22×D7312yes—
265#103—1464no—
266#103—1464no—
267#102—1464no—
268#102—1464no—
269C22⋊C8—24no—
270C4×C8C22⋊D7312yes—
271C4×C8C22×D7312yes—
272C24—21no—
273C2×Q8—2921no—
274C2×Q8—2921no—
275C2×Q8—2922no—
276C2×Q8—2922no—
277C2×Q8—5842no—
278C2×Q8—1462no—
279C2×Q8—1462no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4Dic29214yes—
283C2×D4C2×D29212yes—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—2921no—
290C22×C4—2921no—
291C22×C4C2×C22⋊D7311yescharacteristic
292C22×C4C2×D29211yescharacteristic
293C22×C4#1192402811yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—48no—
298C2×C8—42no—
299C2×C8—2922no—
300C2×C8—2922no—
301C2×C8—1462no—
302C2×C8—1462no—
303C2×C8C2×C22⋊D7312yes—
304C2×C8C2×C22⋊D7312yes—
305C4⋊C4—21no—
306C4⋊C4—2921no—
307C4⋊C4—2921no—
308C4⋊C4—2921no—
309C4⋊C4—2921no—
310C4⋊C4—1462no—
311C4⋊C4—1462no—
312C4⋊C4—1462no—
313C4⋊C4—1462no—
314C4⋊C4C2×D29212yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C42C2×C22⋊D7311yescharacteristic
318C42C2×D29211yescharacteristic
319C42#1192402811yescharacteristic
320C42C23×D7311yescharacteristic
321C23—22no—
322C23—24no—
323C23—42no—
324C23—42no—
325C23#1455656511yescharacteristic
326Q8—2922no—
327Q8—2922no—
328Q8—5842no—
329Q8—5842no—
330Q8—5842no—
331D4—24no—
332D4—22no—
333D4C2×Dic29214yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—2922no—
341C2×C4—2922no—
342C2×C4—5842no—
343C2×C4—5842no—
344C2×C4#1455656511yescharacteristic
345C2×C4#1455656511yescharacteristic
346C2×C4#1455656511yescharacteristic
347C2×C4C2×Dic29212yes—
348C2×C4#1455661112yes—
349C2×C4#1455661312yes—
350C2×C4#1455661712yes—
351C2×C4C22×C22⋊D7311yescharacteristic
352C2×C4C22×D29211yescharacteristic
353C2×C4C2×#1192402811yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—2922no—
358C8—2922no—
359C22—24no—
360C22—42no—
361C22—48no—
362C22—42no—
363C22#1798880212yes—
364C22#1798882812yes—
365C22#1798885012yes—
366C22C2×#1455656511yescharacteristic
367C4—21no—
368C4—21no—
369C4—42no—
370C4—5842no—
371C4—5842no—
372C4#1798880212yes—
373C4#1798882812yes—
374C4C22×Dic29211yescharacteristic
375C4C2×#1455661111yescharacteristic
376C4C2×#1455661311yescharacteristic
377C4C2×#1455661711yescharacteristic
378C2—44no—
379C2#2260938214yesdirect product factor
380C2C2×#1798880211yescharacteristic
381C2C2×#1798882811yescharacteristic
382C2C2×#1798885011yescharacteristic
subgroupG/H
centerC23#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×C8×D4×C73C2
socleC23×C73#121
prime radicalC22×D4×C73D4
hypercenterC2×C8×D4D73
hypocenterC73C2×#1683
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#17988938C2semidirect
4. #22609382C2direct (N × H)
5. C2×#17988340C2semidirect
6. C2×#17988335C2semidirect
7. C2×#17988334C2semidirect
12. #17988938C22semidirect
25. #17988340C22semidirect
26. #17988335C22semidirect
27. #17988334C22semidirect
188. C2×C73#1683semidirect
200. C73C2×#1683semidirect
379. C2#22609382direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179889382
C2×#179888772
C2×#179888672
#226093822
C2×#179883402
C2×#179883352
C2×#179883342
C2×C8×D4×C732
C2×#168373