Finite Group Databasebeta

← #34030709

Subgroups — #34030709

Order 18688.

2858 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989655C212yesmaximal
2#22626446C218yesmaximal, complemented
3Q8×C4⋊#64408C218yesmaximal, complemented
4#22626411C216yesmaximal
5#22626380C216yesmaximal
6C2×C4×Q8×D73C211yesmaximal, characteristic
7#17989655—216no—
8C2×Q8×#64408C22116yes—
9C2×#14556670C22112yes—
10#17989538C22148yes—
11#17989533C22112yes—
12#17989531C22112yes—
13#17989529C2214yes—
14#17989529C22112yes—
15C4×C4⋊#64408C22112yes—
16C4×C4⋊#64408C22112yes—
17C22×Q8×D73C2211yescharacteristic
18C2×Q8×Dic73C411yescharacteristic, complemented
19C2×C4×Dic146C413yescomplemented
20C2×#14556556C413yescomplemented
21C4×Q8×D73C414yescomplemented
22C4×Q8×D73C414yescomplemented
23C4×Q8×D73C2214yes—
24C4×Q8×D73C2214yes—
25C2×C4⋊C4×D73C2213yes—
26C2×C42×D73C2213yes—
27C2×C4×Q8×C73C411yescharacteristic, complemented
28Q8×#64408—264no—
29#14556670—248no—
30C2×C4⋊#64408C23124yes—
31C2×C4⋊#64408C2318yes—
32C2×C4⋊#64408C23124yes—
33C2×C22⋊#64408C2318yes—
34C2×C4×#64408C23124yes—
35(C2×#64408)⋊C4C2318yes—
36(C2×#64408)⋊C4C23124yes—
37C22×Dic146C2×C413yes—
38C2×Q8×D73C2×C414yes—
39C2×Q8×D73D418yes—
40C2×Q8×D73Q818yes—
41C2×Q8×D73C2314yes—
42C22×C4×D73C2311yescharacteristic
43C22×C4×D73C2313yes—
44C22×C4×D73C2313yes—
45Q8×Dic73C2×C414yes—
46Q8×Dic73C2×C414yes—
47C4×Dic146C2×C4112yes—
48C4×Dic146C2×C4112yes—
49#14556556C2×C4112yes—
50#14556556C2×C4112yes—
51C2×C4⋊Dic73C2×C413yes—
52C2×#11924007C2×C416yes—
53C4⋊C4×D73C2×C4112yes—
54C4⋊C4×D73C23112yes—
55C2×C4×Dic73C2×C413yes—
56C2×C4×Dic73C2×C413yes—
57C2×C4×Dic73C2×C413yes—
58C42×D73C2×C416yes—
59C42×D73C2×C416yes—
60C42×D73C2316yes—
61C42×D73C2316yes—
62C22×Q8×C73C2×C411yescharacteristic
63C4×Q8×C73C2×C414yes—
64C4×Q8×C73C2×C414yes—
65C2×C4⋊C4×C73C2×C413yes—
66C2×C42×C73C2×C413yes—
67C22×#64408C24116yes—
68C4⋊#64408—248no—
69C4⋊#64408—216no—
70C4⋊#64408—424no—
71C22⋊#64408—216no—
72C4×#64408—248no—
73C23×D73C2411yescharacteristic
74C2×Dic146C4⋊C4124yescomplemented
75C2×Dic146C22×C4112yes—
76Q8×D73C4⋊C4132yescomplemented
77Q8×D73C2×D4116yes—
78Q8×D73C2×Q8116yes—
79C22×Dic73C22×C411yescharacteristic
80C22×Dic73C22×C413yes—
81C22×Dic73C22×C413yes—
82C2×C4×D73—24no—
83C2×C4×D73—46no—
84C2×C4×D73C22×C411yescharacteristic
85C2×C4×D73C22×C411yescharacteristic
86C2×C4×D73C22×C416yes—
87C2×C4×D73C22×C416yes—
88C2×C4×D73C2×D4112yes—
89C2×C4×D73C2×Q8112yes—
90C2×C4×D73C2411yescharacteristic
91C2×C4×D73C2416yes—
92C2×C4×D73C2416yes—
93C2×C4×D73C2411yescharacteristic
94C4⋊Dic73C22×C4112yes—
95#11924007C22×C4124yes—
96C4×Dic73C22×C416yes—
97C4×Dic73C22×C416yes—
98C4×Dic73C22×C416yes—
99C4×Dic73C22×C416yes—
100C4×Dic73C22×C416yes—
101C4×Dic73C22×C416yes—
102C2×Q8×C73C4⋊C418yescomplemented
103C2×Q8×C73C22×C414yes—
104C22×C4×C73C22×C411yescharacteristic
105C22×C4×C73C22×C413yes—
106C22×C4×C73C22×C413yes—
107C4⋊C4×C73C22×C4112yes—
108C42×C73C22×C416yes—
109C42×C73C22×C416yes—
110C23×C73C23×C411yescharacteristic
111C22×D73—22no—
112C22×D73—22no—
113C22×D73C23×C411yescharacteristic
114C22×D73C22×D412yes—
115C22×D73C22×Q812yes—
116C22×D73C2511yescharacteristic
117C2×#64408—216no—
118C2×#64408—216no—
119C2×#64408—216no—
120Q8×C73C2×C4⋊C4116yes—
121C2×C4×C73—22no—
122C2×C4×C73—43no—
123C2×C4×C73C2×C4⋊C4112yes—
124C2×C4×C73C23×C411yescharacteristic
125C2×C4×C73C23×C416yes—
126C2×C4×C73C23×C416yes—
127C2×C4×C73C23×C411yescharacteristic
128C2×Dic73—22no—
129C2×Dic73—43no—
130C2×Dic73C2×C4⋊C4112yes—
131C2×Dic73C23×C411yescharacteristic
132C2×Dic73C23×C416yes—
133C2×Dic73C23×C416yes—
134C2×Dic73C23×C411yescharacteristic
135C4×D73—22no—
136C4×D73—212no—
137C4×D73—22no—
138C4×D73—22no—
139C4×D73—46no—
140C4×D73C2×C4⋊C4124yes—
141C4×D73C22×D4112yes—
142C4×D73C22×Q8112yes—
143C4×D732−1+414yes—
144Dic146C2×C4⋊C4148yes—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73C22×C4⋊C412yes—
148C22×C73C24×C411yescharacteristic
149C2×D73—21no—
150C2×D73—21no—
151C2×D73—24no—
152C2×D73—24no—
153C2×D73C22×C4⋊C412yes—
154C2×D73C23×D411yescharacteristic
155C2×D73C23×Q811yescharacteristic
156C2×D73C2×2−1+412yes—
157C2×D73#58512yes—
158#64408—416no—
159C4×C73—21no—
160C4×C73—26no—
161C4×C73C22×C4⋊C4112yes—
162C4×C73#51912yescomplemented
163Dic73—21no—
164Dic73—26no—
165Dic73C22×C4⋊C4112yes—
166Dic73#51912yescomplemented
167#34182—731nomaximal, Sylow
168C2×C73—22no—
169C2×C73C23×C4⋊C411yescharacteristic
170C2×C73C2×#51911yescharacteristic
171C2×C73#342811yescharacteristic
172D73—22no—
173D73#349712yes—
174D73#350212yes—
175C2×#519—732no—
176C22×C4×Q8—731no—
177#2351—738no—
178Q8×C4⋊C4—738no—
179#2316—736no—
180#2285—736no—
181C73#3418211yescharacteristic, complemented, Sylow
182C23×Q8—731no—
183#519—14616no—
184C2×C4×Q8—731no—
185C2×C4×Q8—734no—
186C2×C4×Q8—734no—
187C2×C4×Q8—734no—
188C2×C4×Q8—734no—
189C2×C4×Q8—7316no—
190C2×C4×Q8—733no—
191C2×C4×Q8—733no—
192C2×C4×Q8#6440811yescharacteristic, complemented
193C2×#117—7312no—
194C22×C4⋊C4—733no—
195C22×C42—733no—
196#389—7348no—
197#384—7312no—
198#382—7312no—
199#380—7312no—
200#380—734no—
201C4×C4⋊C4—7312no—
202C4×C4⋊C4—7312no—
203C22×Q8—738no—
204C22×Q8—738no—
205C22×Q8—734no—
206C22×Q8—734no—
207C22×Q8—733no—
208C22×Q8C2×#6440811yescharacteristic
209C23×C4—733no—
210C23×C4—733no—
211C23×C4—731no—
212C4×Q8—734no—
213C4×Q8—734no—
214C4×Q8—7312no—
215C4×Q8—7312no—
216C4×Q8—7312no—
217C4×Q8—7312no—
218C4×Q8—14664no—
219C4×Q8C2×#6440814yes—
220C4×Q8C2×#6440814yes—
221#117—14648no—
222C2×C4⋊C4—7324no—
223C2×C4⋊C4—736no—
224C2×C4⋊C4—733no—
225C2×C4⋊C4—7312no—
226C2×C4⋊C4—7312no—
227C2×C4⋊C4—7324no—
228C2×C4⋊C4—738no—
229C2×C4⋊C4C2×#6440813yes—
230C2×C22⋊C4—738no—
231C2×C42—733no—
232C2×C42—733no—
233C2×C42—733no—
234C2×C42—736no—
235C2×C42—736no—
236C2×C42—736no—
237C2×C42—736no—
238C2×C42—7324no—
239C2×C42C2×#6440813yes—
240#95—738no—
241#95—7324no—
242C24—731no—
243C2×Q8—7332no—
244C2×Q8—7316no—
245C2×Q8—7316no—
246C2×Q8—7312no—
247C2×Q8—7324no—
248C2×Q8C4⋊#6440818yescomplemented
249C2×Q8C22×#6440814yes—
250C22×C4—2926no—
251C22×C4—733no—
252C22×C4—736no—
253C22×C4—733no—
254C22×C4—736no—
255C22×C4—731no—
256C22×C4—736no—
257C22×C4—7312no—
258C22×C4—7312no—
259C22×C4—731no—
260C22×C4—731no—
261C22×C4—731no—
262C22×C4—731no—
263C22×C4—736no—
264C22×C4—7316no—
265C22×C4—1464no—
266C22×C4C22×#6440811yescharacteristic
267C22×C4C22×#6440813yes—
268C22×C4C22×#6440813yes—
269C4⋊C4—29224no—
270C4⋊C4—7324no—
271C4⋊C4—7312no—
272C4⋊C4—14616no—
273C4⋊C4—14648no—
274C4⋊C4C22×#64408112yes—
275C22⋊C4—14616no—
276C42—736no—
277C42—736no—
278C42—736no—
279C42—736no—
280C42—736no—
281C42—736no—
282C42—14648no—
283C42C22×#6440816yes—
284C42C22×#6440816yes—
285C23—731no—
286C23—732no—
287C23—732no—
288C23—731no—
289C23—1462no—
290C23—1462no—
291C23C23×#6440811yescharacteristic
292Q8—7348no—
293Q8C2×C4⋊#64408116yes—
294C2×C4—22no—
295C2×C4—43no—
296C2×C4—2926no—
297C2×C4—2923no—
298C2×C4—7312no—
299C2×C4—731no—
300C2×C4—736no—
301C2×C4—7312no—
302C2×C4—7324no—
303C2×C4—7312no—
304C2×C4—734no—
305C2×C4—731no—
306C2×C4—736no—
307C2×C4—1462no—
308C2×C4—14612no—
309C2×C4—1462no—
310C2×C4—14616no—
311C2×C4—14616no—
312C2×C4—14616no—
313C2×C4—1462no—
314C2×C4—1462no—
315C2×C4C2×C4⋊#64408112yes—
316C2×C4C23×#6440811yescharacteristic
317C2×C4C23×#6440816yes—
318C2×C4C23×#6440816yes—
319C2×C4C23×#6440811yescharacteristic
320C22—21no—
321C22—21no—
322C22—731no—
323C22—731no—
324C22—732no—
325C22—732no—
326C22—732no—
327C22—1464no—
328C22—1464no—
329C22—1461no—
330C22—1461no—
331C22C22×C4⋊#6440812yes—
332C22C24×#6440811yescharacteristic
333C4—21no—
334C4—26no—
335C4—29216no—
336C4—7312no—
337C4—732no—
338C4—1461no—
339C4—1466no—
340C4C22×C4⋊#64408112yes—
341C4#1798965512yescomplemented
342C2—22no—
343C2—732no—
344C2—732no—
345C2—1462no—
346C2C23×C4⋊#6440811yescharacteristic
347C2C2×#1798965511yescharacteristic
348C2#2262691111yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
FittingC2×C4×Q8×C73C4
prime radicalC23×D73C24
cosocleC22×D73C25
hypercenterC2×C4×Q8#64408
hypocenterC73#34182
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22626446C2semidirect
3. Q8×C4⋊#64408C2semidirect
18. C2×Q8×Dic73C4semidirect
19. C2×C4×Dic146C4semidirect
20. C2×#14556556C4semidirect
21. C4×Q8×D73C4semidirect
22. C4×Q8×D73C4semidirect
27. C2×C4×Q8×C73C4semidirect
74. C2×Dic146C4⋊C4semidirect
76. Q8×D73C4⋊C4semidirect
102. C2×Q8×C73C4⋊C4semidirect
162. C4×C73#519semidirect
166. Dic73#519semidirect
181. C73#34182semidirect
192. C2×C4×Q8#64408semidirect
248. C2×Q8C4⋊#64408semidirect
341. C4#17989655semidirect

Maximal subgroups

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

subgroupindex
C2×#179896552
#226264462
Q8×C4⋊#644082
#226264112
#226263802
C2×C4×Q8×D732
#3418273