Finite Group Databasebeta

← #33243708

Subgroups — #33243708

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×D584C211yesmaximal, characteristic, complemented
2#22612167C212yesmaximal, complemented
3#22612158C211yesmaximal, characteristic, complemented
4#22612083C212yesmaximal, complemented
5#22612068C211yesmaximal, characteristic, complemented
6#22611983C212yesmaximal
7#22611973C211yesmaximal, characteristic, complemented
8#22611840C211yesmaximal, characteristic, complemented
9#22611793C212yesmaximal, complemented
10C4×#14556532C211yesmaximal, characteristic
11C73×#3393C211yesmaximal, characteristic, complemented
12Q8×D292—21no—
13C2×#14556534C2212yes—
14C2×#14556532C2211yescharacteristic
15#17988992C2211yescharacteristic
16#17988992C2212yes—
17#17988868—21no—
18#17988868—22no—
19C4×D584C2211yescharacteristic
20C4×D584C2212yes—
21#17988412C2211yescharacteristic
22#17988412C2212yes—
23C4⋊#11924003C2211yescharacteristic
24C4⋊#11924003C2212yes—
25#17988391C2212yes—
26#17988391C2212yes—
27#17988391C2212yes—
28#17988388C2212yes—
29#17988388C2211yescharacteristic
30#17988387C2211yescharacteristic
31#17988363C2212yes—
32C4×#11924003C2211yescharacteristic
33#556×C73—21no—
34#495×C73C2212yes—
35#494×C73C2211yescharacteristic
36#464×C73—22no—
37#461×C73—21no—
38C8×Q8×C73C2211yescharacteristic
39C4×Q16×C73C2212yes—
40C4×D8×C73C2211yescharacteristic
41C2×Q8×D73—41no—
42C2×D584C2311yescharacteristic
43#14556591—21no—
44#14556591—22no—
45#14556571—41no—
46#14556571—42no—
47C4×D292—21no—
48C4×D292—22no—
49#14556534—24no—
50#14556532—22no—
51C2×#11924003C2311yescharacteristic
52C2×#11924003C2311yescharacteristic
53C2×#11924003C2312yes—
54#14556525—21no—
55#14556525—22no—
56#14556525C2312yes—
57#14556525C2311yescharacteristic
58#14556513—21no—
59#14556513—22no—
60C2×Q16×C73—21no—
61C2×Q16×C73C2312yes—
62C2×SD16×C73—22no—
63C2×D8×C73C2311yescharacteristic
64C4⋊Q8×C73—21no—
65#126×C73—42no—
66#124×C73—22no—
67C22⋊Q8×C73—41no—
68C4×Q8×C73—22no—
69C4×Q8×C73D411yescharacteristic
70C4×D4×C73—21no—
71#117×C73—41no—
72#107×C73C2311yescharacteristic
73C4⋊C8×C73D411yescharacteristic
74C4⋊C8×C73D412yes—
75#103×C73—22no—
76#102×C73—21no—
77C4×C8×C73C2311yescharacteristic
78C4×C8×C73C2312yes—
79C2×Dic146—42no—
80C2×Dic146—41no—
81Q8×D73—84no—
82C2×D292—21no—
83C2×C4×D73—42no—
84C2×C4×D73—41no—
85D584—21no—
86D584#4112yes—
87C4⋊Dic73—21no—
88C4⋊Dic73—22no—
89C4⋊Dic73—42no—
90C4⋊Dic73—41no—
91#11924005—42no—
92#11924005—41no—
93#11924003—21no—
94#11924003—22no—
95#11924003—21no—
96C2×Q8×C73—22no—
97C2×Q8×C73D811yescharacteristic
98C2×D4×C73—21no—
99C22×C4×C73—41no—
100Q16×C73—22no—
101Q16×C73—22no—
102Q16×C73—81no—
103SD16×C73—82no—
104D8×C73—21no—
105D8×C73—21no—
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D412yes—
108C2×C8×C73C2411yescharacteristic
109C4⋊C4×C73—21no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—42no—
112C4⋊C4×C73—41no—
113C4⋊C4×C73D811yescharacteristic
114C4⋊C4×C73D812yes—
115C22⋊C4×C73—42no—
116C22⋊C4×C73—41no—
117C22⋊C4×C73—42no—
118C42×C73—42no—
119C42×C73C2×D411yescharacteristic
120C42×C73C2×D412yes—
121C23×C73—41no—
122C22×D73—41no—
123Q8×C73—22no—
124Q8×C73—42no—
125Q8×C73—42no—
126D4×C73—41no—
127D4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—42no—
130C2×C4×C73—82no—
131C2×C4×C73C2×D812yes—
132C2×C4×C73C2×D811yescharacteristic
133C2×C4×C73C2×D812yes—
134C2×C4×C73C2×D811yescharacteristic
135C2×C4×C73C22×D411yescharacteristic
136C2×Dic73—41no—
137C2×Dic73—42no—
138D292—22no—
139D292—41no—
140C4×D73—84no—
141C4×D73—82no—
142Dic146—84no—
143Dic146—82no—
144C8×C73—22no—
145C8×C73—21no—
146C8×C73C2×#4111yescharacteristic
147C8×C732−1+411yescharacteristic
148C22×C73—81no—
149C22×C73—81no—
150C22×C73—81no—
151C22×C73C22×D811yescharacteristic
152C2×D73—42no—
153C2×D73—81no—
154C2×D73—81no—
155C4×C73—22no—
156C4×C73—22no—
157C4×C73—21no—
158C4×C73—82no—
159C4×C73—81no—
160C4×C73#50912yes—
161C4×C73#54811yescharacteristic
162C4×C73#57711yescharacteristic
163Dic73—81no—
164Dic73—82no—
165#33934—731nomaximal, Sylow
166C2×C73—161no—
167C2×C73#335811yescharacteristic
168C2×C73C2×#50911yescharacteristic
169C2×C73#341711yescharacteristic
170D73—82no—
171#3393D7311yescharacteristic, complemented
172Q8×D8—731no—
173#2243—732no—
174#2241—731no—
175#2227—732no—
176#2224—731no—
177#2210—732no—
178#2208—731no—
179#2183—731no—
180#2175—732no—
181C4×D16—731no—
182C73#3393411yescharacteristic, complemented, Sylow
183#556—21no—
184D4×Q8—1461no—
185C2×SD32—732no—
186C2×D16—731no—
187#500—731no—
188#500—732no—
189#495C2×D7312yes—
190#494C2×D7311yescharacteristic
191#474—1461no—
192#474—1462no—
193#464—22no—
194#461—21no—
195C8×Q8C2×D7311yescharacteristic
196C4×Q16C2×D7312yes—
197C4×D8—731no—
198C4×D8—732no—
199C4×D8C2×D7311yescharacteristic
200#366—731no—
201#366—732no—
202C4⋊C16—731no—
203C4⋊C16—732no—
204#358—732no—
205#358—732no—
206#358—732no—
207#357—731no—
208#357—732no—
209#357—731no—
210#346—732no—
211C4×C16—731no—
212C22×Q8—2921no—
213C2×Q16—21no—
214C2×Q16C22×D7312yes—
215C2×SD16—22no—
216C2×D8—731no—
217C2×D8C22×D7311yescharacteristic
218C4⋊Q8—21no—
219C4⋊Q8—1461no—
220C4⋊Q8—1462no—
221#126—42no—
222#124—22no—
223C22⋊Q8—41no—
224C22⋊Q8—2921no—
225C22⋊Q8—2922no—
226C4×Q8—22no—
227C4×Q8C22⋊D7311yescharacteristic
228C4×D4—21no—
229C4×D4—1461no—
230C4×D4—1462no—
231#117—41no—
232SD32—1464no—
233D16—1462no—
234C2×C16—731no—
235C2×C16—731no—
236C2×C16—732no—
237#107—731no—
238#107—732no—
239#107—1461no—
240#107—1462no—
241#107C22×D7311yescharacteristic
242C4⋊C8C22⋊D7311yescharacteristic
243C4⋊C8C22⋊D7312yes—
244#103—22no—
245#102—21no—
246#102—1461no—
247#102—1462no—
248C4×C8C22×D7311yescharacteristic
249C4×C8C22×D7312yes—
250C2×Q8—22no—
251C2×Q8—2922no—
252C2×Q8—2921no—
253C2×Q8—5844no—
254C2×Q8#1192401311yescharacteristic
255C2×D4—21no—
256C2×D4—1461no—
257C22×C4—41no—
258C22×C4—2922no—
259C22×C4—2921no—
260Q16—22no—
261Q16—22no—
262Q16—81no—
263SD16—82no—
264D8—21no—
265D8—21no—
266D8—732no—
267D8—1461no—
268C2×C8C2×C22⋊D7311yescharacteristic
269C2×C8C2×C22⋊D7312yes—
270C2×C8C23×D7311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—2922no—
276C4⋊C4—2921no—
277C4⋊C4—1461no—
278C4⋊C4—1462no—
279C4⋊C4#1192401311yescharacteristic
280C4⋊C4#1192401312yes—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—42no—
284C22⋊C4—2922no—
285C22⋊C4—2921no—
286C42—42no—
287C42C2×C22⋊D7311yescharacteristic
288C42C2×C22⋊D7312yes—
289C16—1461no—
290C16—1462no—
291C16—1461no—
292C23—41no—
293C23—2921no—
294Q8—22no—
295Q8—42no—
296Q8—42no—
297Q8—5844no—
298Q8—5842no—
299D4—41no—
300D4—41no—
301D4—2921no—
302D4—1462no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—2922no—
307C2×C4—2921no—
308C2×C4—5844no—
309C2×C4—5842no—
310C2×C4C2×#1192401312yes—
311C2×C4C2×#1192401311yescharacteristic
312C2×C4C2×#1192401312yes—
313C2×C4C2×#1192401311yescharacteristic
314C2×C4C22×C22⋊D7311yescharacteristic
315C8—22no—
316C8—21no—
317C8#41×D7311yescharacteristic
318C8#1455663811yescharacteristic
319C22—81no—
320C22—81no—
321C22—81no—
322C22—2922no—
323C22—5841no—
324C22—5841no—
325C22C22×#1192401311yescharacteristic
326C4—22no—
327C4—22no—
328C4—21no—
329C4—81no—
330C4—82no—
331C4—5842no—
332C4—5841no—
333C4#1798902212yes—
334C4#1798924411yescharacteristic
335C4#1798939811yescharacteristic
336C2—161no—
337C2—5842no—
338C2#2262256211yescharacteristic
339C2C2×#1798902211yescharacteristic
340C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3393C2
socleC22×C73C22×D8
prime radicalC2×#14556532C22
hypercenter#3393D73
hypocenterC73#33934
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. Q8×D584C2semidirect
2. #22612167C2semidirect
3. #22612158C2semidirect
4. #22612083C2semidirect
5. #22612068C2semidirect
7. #22611973C2semidirect
8. #22611840C2semidirect
9. #22611793C2semidirect
11. C73×#3393C2semidirect
171. #3393D73semidirect
182. C73#33934semidirect

Maximal subgroups

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

subgroupindex
Q8×D5842
#226121672
#226121582
#226120832
#226120682
#226119832
#226119732
#226118402
#226117932
C4×#145565322
C73×#33932
#3393473