Finite Group Databasebeta

← #33003048

Subgroups — #33003048

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556534C211yesmaximal, characteristic
2C2×#17988391C211yesmaximal, characteristic, complemented
3C2×#17988387C211yesmaximal, characteristic
4#22610797C211yesmaximal, characteristic
5#22608022C211yesmaximal, characteristic, complemented
6#22608009C211yesmaximal, characteristic, complemented
7C2×C4×Q16×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C2×#14556534—22no—
10C2×#14556534—22no—
11C2×#14556534C414yescomplemented
12C22×#11924003C2211yescharacteristic
13C22×#11924003C2211yescharacteristic
14C2×#14556525C2211yescharacteristic
15C2×#14556513—21no—
16#17988489—21no—
17#17988391—22no—
18#17988387—22no—
19C22×Q16×C73C2211yescharacteristic
20C2×C4×Q8×C73—21no—
21C4×Q16×C73—22no—
22C4×Q16×C73—22no—
23C2×#107×C73C2211yescharacteristic
24C2×#103×C73—21no—
25C2×C4×C8×C73C2211yescharacteristic
26C22×D292—21no—
27C2×D584—21no—
28C2×D584—21no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584C2×C412yes—
33C2×C4⋊Dic73—21no—
34C2×#11924005—41no—
35C2×#11924005—41no—
36#14556534—28no—
37C2×#11924003—21no—
38C2×#11924003—21no—
39C2×#11924003—21no—
40C2×#11924003—21no—
41C2×#11924003—21no—
42C2×#11924003C2×C412yes—
43#14556525—21no—
44#14556525—21no—
45#14556513—42no—
46C22×Q8×C73—21no—
47C2×Q16×C73—21no—
48C2×Q16×C73—21no—
49C2×Q16×C73—21no—
50C2×Q16×C73C2×C412yes—
51C2×Q16×C73D412yes—
52C2×Q16×C73D412yes—
53C22×C8×C73D411yescharacteristic
54C22×C8×C73C2311yescharacteristic
55C4×Q8×C73—24no—
56C4×Q8×C73—42no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—41no—
59C2×C42×C73—41no—
60C2×C42×C73D411yescharacteristic
61#107×C73—21no—
62#107×C73D412yes—
63#103×C73—24no—
64C4×C8×C73—21no—
65C4×C8×C73—21no—
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C22×Dic73—41no—
74D584—22no—
75D584—22no—
76D584—41no—
77D584—41no—
78C4⋊Dic73—41no—
79C4⋊Dic73—41no—
80#11924005—82no—
81#11924005—82no—
82#11924003—22no—
83#11924003—41no—
84C2×Q8×C73—22no—
85C2×Q8×C73—22no—
86C2×Q8×C73—22no—
87C2×Q8×C73—41no—
88C2×Q8×C73—41no—
89C2×Q8×C73—41no—
90C22×C4×C73—41no—
91C22×C4×C73—41no—
92C22×C4×C73D811yescharacteristic
93C22×C4×C73D811yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95Q16×C73—22no—
96Q16×C73—22no—
97Q16×C73—22no—
98Q16×C73C22⋊C414yescomplemented
99C2×C8×C73—21no—
100C2×C8×C73—21no—
101C2×C8×C73—21no—
102C2×C8×C73C22×C411yescharacteristic
103C2×C8×C73C2×D411yescharacteristic
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73#4111yescharacteristic
107C2×C8×C73#4111yescharacteristic
108C4⋊C4×C73—22no—
109C4⋊C4×C73—44no—
110C4⋊C4×C73—41no—
111C42×C73—21no—
112C42×C73—21no—
113C42×C73—44no—
114C23×C73C2×D811yescharacteristic
115C22×D73—42no—
116C22×D73—81no—
117C22×D73—81no—
118C22×D73—81no—
119C22×D73—81no—
120C22×D73—81no—
121C22×D73—81no—
122Q8×C73—24no—
123Q8×C73—42no—
124Q8×C73—42no—
125Q8×C73—42no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73—81no—
136C2×C4×C73—81no—
137C2×C4×C73SD3212yescomplemented
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73#12011yescharacteristic
140C2×C4×C73#12111yescharacteristic
141C2×C4×C73#13511yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C2×Dic73—81no—
145C2×Dic73—81no—
146C2×Dic73—81no—
147D292—42no—
148D292—42no—
149D292—81no—
150D292—81no—
151C8×C73—41no—
152C8×C73C2×C22⋊C411yescharacteristic
153C8×C73C4×D411yescharacteristic
154C8×C73#12111yescharacteristic
155C8×C73#12311yescharacteristic
156C22×C73C4×D811yescharacteristic
157C22×C73#44711yescharacteristic
158C22×C73#45911yescharacteristic
159C22×C73C2×SD3211yescharacteristic
160C22×C73#50811yescharacteristic
161C22×C73#50911yescharacteristic
162C22×C73#51011yescharacteristic
163C2×D73—82no—
164C2×D73—82no—
165C2×D73—82no—
166C2×D73—161no—
167C2×D73—161no—
168C2×D73—161no—
169C2×D73—161no—
170C4×C73—22no—
171C4×C73—44no—
172C4×C73—41no—
173C4×C73—82no—
174C4×C73#38611yescharacteristic
175C4×C73#44211yescharacteristic
176C4×C73#44911yescharacteristic
177C4×C73#46311yescharacteristic
178Dic73—161no—
179#13013—731nomaximal, Sylow
180C2×C73#187911yescharacteristic
181C2×C73C4×SD3211yescharacteristic
182C2×C73#217511yescharacteristic
183C2×C73#218511yescharacteristic
184C2×C73#218611yescharacteristic
185C2×C73#220711yescharacteristic
186C2×C73#221111yescharacteristic
187D73—162no—
188C22×SD32—731no—
189C2×C4×Q16D7311yescharacteristic, complemented
190C2×#358—731no—
191C2×#357—731no—
192#1970—731no—
193#1380—731no—
194#1374—731no—
195C73#1301311yescharacteristic, complemented, Sylow
196C22×Q16C2×D7311yescharacteristic
197C22×D8—731no—
198C2×C4×Q8—21no—
199C2×SD32—734no—
200C2×SD32—1462no—
201C2×SD32—1462no—
202C22×C16—731no—
203C22×C16—731no—
204C4×Q16—22no—
205C4×Q16—22no—
206C2×#107—731no—
207C2×#107C2×D7311yescharacteristic
208C2×#103—21no—
209C2×#102—1461no—
210C2×C4×C8C2×D7311yescharacteristic
211#390—1461no—
212#358—1462no—
213#357—1462no—
214C22×Q8—21no—
215C22×D4—1461no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D7312yes—
220C2×Q16D29212yes—
221C2×Q16C22⋊D7312yes—
222C2×D8—732no—
223C2×D8—1461no—
224C2×D8—1461no—
225C2×D8—1461no—
226C2×D8—1461no—
227C2×D8—1461no—
228C22×C8C22⋊D7311yescharacteristic
229C22×C8C22×D7311yescharacteristic
230C4×Q8—24no—
231C4×Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—1461no—
235C2×C22⋊C4—2921no—
236C2×C22⋊C4—2921no—
237C2×C42—41no—
238C2×C42C22⋊D7311yescharacteristic
239SD32—1468no—
240C2×C16—732no—
241C2×C16—1461no—
242C2×C16—1461no—
243C2×C16—1461no—
244C2×C16—1461no—
245C2×C16—1461no—
246#107—21no—
247#107—1461no—
248#107—1461no—
249#107C22⋊D7312yes—
250#103—24no—
251#102—2922no—
252C4×C8—21no—
253C4×C8—21no—
254C24—2921no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—2921no—
262C2×D4—2921no—
263C2×D4—2921no—
264C2×D4—2921no—
265C2×D4—2921no—
266C2×D4—1462no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—2921no—
270C22×C4#1192401311yescharacteristic
271C22×C4#1192401311yescharacteristic
272C22×C4C2×C22⋊D7311yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16#1192400514yescomplemented
277D8—2921no—
278D8—2921no—
279D8—1462no—
280D8—1462no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D7311yescharacteristic
285C2×C8C2×C22⋊D7311yescharacteristic
286C2×C8C2×C22⋊D7311yescharacteristic
287C2×C8C2×D29211yescharacteristic
288C2×C8#1192402811yescharacteristic
289C2×C8#1192402811yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—2921no—
294C4⋊C4—2921no—
295C22⋊C4—5842no—
296C22⋊C4—5842no—
297C42—21no—
298C42—21no—
299C42—44no—
300C16—2921no—
301C16—1462no—
302C23—2922no—
303C23—5841no—
304C23—5841no—
305C23—5841no—
306C23—5841no—
307C23—5841no—
308C23—5841no—
309C23C2×#1192401311yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—2922no—
315D4—2922no—
316D4—5841no—
317D4—5841no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—5841no—
330C2×C4—5841no—
331C2×C4—5841no—
332C2×C4#1455653412yescomplemented
333C2×C4C4×C22⋊D7311yescharacteristic
334C2×C4#1455656011yescharacteristic
335C2×C4#1455656511yescharacteristic
336C2×C4#1455661111yescharacteristic
337C2×C4#1455661611yescharacteristic
338C2×C4#1455661711yescharacteristic
339C8—41no—
340C8C2×#1192400511yescharacteristic
341C8C4×C22⋊D7311yescharacteristic
342C8#1455656511yescharacteristic
343C8#1455657411yescharacteristic
344C22—5842no—
345C22—5842no—
346C22—5842no—
347C22—11681no—
348C22—11681no—
349C22—11681no—
350C22—11681no—
351C22C4×#1192401311yescharacteristic
352C22#1798872111yescharacteristic
353C22#1798878211yescharacteristic
354C22C2×#1455653411yescharacteristic
355C22#1798901711yescharacteristic
356C22#1798902211yescharacteristic
357C22#1798902811yescharacteristic
358C4—22no—
359C4—44no—
360C4—41no—
361C4—82no—
362C4—11681no—
363C4#1798846611yescharacteristic
364C4#1798869711yescharacteristic
365C4#1798873211yescharacteristic
366C4#1798880211yescharacteristic
367C2—11682no—
368C2#2261027911yescharacteristic
369C2C4×#1455653411yescharacteristic
370C2#2261179311yescharacteristic
371C2#2261184811yescharacteristic
372C2#2261185411yescharacteristic
373C2#2261197111yescharacteristic
374C2#2261199311yescharacteristic
subgroupG/H
centerC23C2×#11924013
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×C4×Q16×C73C2
socleC23×C73C2×D8
prime radicalC22×D584C22
cosocleC22×C8×C73C23
hypercenterC2×C4×Q16D73
hypocenterC73#13013
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#17988391C2semidirect
5. #22608022C2semidirect
6. #22608009C2semidirect
7. C2×C4×Q16×C73C2semidirect
11. C2×#14556534C4semidirect
98. Q16×C73C22⋊C4semidirect
137. C2×C4×C73SD32semidirect
189. C2×C4×Q16D73semidirect
195. C73#13013semidirect
276. Q16#11924005semidirect
332. C2×C4#14556534semidirect

Maximal subgroups

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

subgroupindex
C22×#145565342
C2×#179883912
C2×#179883872
#226107972
#226080222
#226080092
C2×C4×Q16×C732
#1301373