Finite Group Databasebeta

← #34002890

Subgroups — #34002890

Order 18688.

790 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989605C211yesmaximal, characteristic
2C2×#17989592C211yesmaximal, characteristic
3#22624809C214yesmaximal, complemented
4#384×D73C211yesmaximal, characteristic
5C22×#11924038—21no—
6#17989605—22no—
7#17989605C414yescomplemented
8#17989592—22no—
9#17989592C414yescomplemented
10C2×#14556658—21no—
11C2×#14556658C2212yes—
12C2×#14556658C2212yes—
13C2×#14556648—21no—
14C2×C4×#10596833—21no—
15C2×C4⋊C4×D73—21no—
16C2×C4⋊C4×D73—21no—
17C2×C4⋊C4×D73C412yes—
18C2×C42×D73C2211yescharacteristic
19#17988457C411yescharacteristic
20#17988456—21no—
21#17988456—21no—
22#17988456C2212yes—
23#384×C73C411yescharacteristic
24C2×#11924038—24no—
25C2×#11924038—42no—
26C2×#11924038—42no—
27C22×#10596833—22no—
28C22×#10596833—22no—
29#14556658—24no—
30#14556658—22no—
31#14556658—22no—
32#14556658C2×C414yescomplemented
33#14556658C2×C414yescomplemented
34#14556648—24no—
35C4×#10596833—24no—
36C22×C4×D73—22no—
37C22×C4×D73C2×C411yescharacteristic
38C22×C4×D73D411yescharacteristic
39C22×C4×D73D411yescharacteristic
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73—21no—
42C2×C4⋊Dic73C2×C412yes—
43C2×#11924007—22no—
44C2×#11924007—22no—
45C2×#11924007—22no—
46C4⋊C4×D73—44no—
47C4⋊C4×D73—42no—
48C4⋊C4×D73—42no—
49C4⋊C4×D73—44no—
50C2×C4×Dic73D411yescharacteristic
51C2×C4×Dic73D411yescharacteristic
52C2×C4×Dic73C2311yescharacteristic
53C42×D73C2×C412yes—
54C42×D73C2×C412yes—
55C42×D73D412yes—
56C42×D73Q812yes—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73C2×C412yes—
60C2×C42×C73C2×C411yescharacteristic
61#11924038—48no—
62C2×#10596833—24no—
63C2×#10596833—24no—
64C2×#10596833—42no—
65C2×#10596833—42no—
66C2×#10596833—42no—
67C2×#10596833—42no—
68C23×D73C22⋊C411yescharacteristic
69C22×Dic73—22no—
70C22×Dic73C22⋊C411yescharacteristic
71C22×Dic73C2×D411yescharacteristic
72C22×Dic73C2×D411yescharacteristic
73C2×C4×D73—22no—
74C2×C4×D73—22no—
75C2×C4×D73—22no—
76C2×C4×D73—22no—
77C2×C4×D73—44no—
78C2×C4×D73—44no—
79C2×C4×D73—44no—
80C2×C4×D73C4212yes—
81C2×C4×D73C22⋊C412yes—
82C2×C4×D73C22⋊C412yes—
83C2×C4×D73C4⋊C412yes—
84C2×C4×D73C4⋊C412yes—
85C2×C4×D73C4⋊C412yes—
86C2×C4×D73D812yes—
87C2×C4×D73SD1612yes—
88C2×C4×D73SD1612yes—
89C2×C4×D73Q1612yes—
90C4⋊Dic73—42no—
91C4⋊Dic73—42no—
92C4⋊Dic73—42no—
93#11924007—44no—
94#11924007—42no—
95#11924007—42no—
96#11924007—44no—
97C4×Dic73C22⋊C412yes—
98C4×Dic73C22⋊C412yes—
99C4×Dic73C4⋊C412yes—
100C4×Dic73C4⋊C412yes—
101C4×Dic73C22×C411yescharacteristic
102C4×Dic73C22×C411yescharacteristic
103C4×Dic73C2×D411yescharacteristic
104C4×Dic73C2×Q811yescharacteristic
105C22×C4×C73—22no—
106C22×C4×C73C22⋊C411yescharacteristic
107C22×C4×C73C22⋊C411yescharacteristic
108C22×C4×C73C22×C411yescharacteristic
109C4⋊C4×C73—42no—
110C4⋊C4×C73—42no—
111C4⋊C4×C73—42no—
112C42×C73C4212yes—
113C42×C73C4⋊C412yes—
114C23×C73C2×C22⋊C411yescharacteristic
115C22×D73—21no—
116C22×D73—21no—
117C22×D73—21no—
118C22×D73—21no—
119C22×D73#9512yes—
120C22×D73#10212yes—
121C22×D73#10312yes—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—42no—
129C2×C4×C73#9512yes—
130C2×C4×C73#9512yes—
131C2×C4×C73#10212yes—
132C2×C4×C73#10312yes—
133C2×C4×C73C2×C4211yescharacteristic
134C2×C4×C73C2×C4⋊C411yescharacteristic
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—42no—
138C2×Dic73#9512yes—
139C2×Dic73#10012yes—
140C2×Dic73#10112yes—
141C2×Dic73C2×C22⋊C411yescharacteristic
142C2×Dic73C2×C22⋊C411yescharacteristic
143C2×Dic73C2×C4⋊C411yescharacteristic
144C2×Dic73C2×C4⋊C411yescharacteristic
145C2×Dic73C2×D811yescharacteristic
146C2×Dic73C2×SD1611yescharacteristic
147C2×Dic73C2×SD1611yescharacteristic
148C2×Dic73C2×Q1611yescharacteristic
149C2×Dic73AGL(1,ℤ/8)12yes—
150C2×Dic73#13712yes—
151C4×D73—24no—
152C4×D73—24no—
153C4×D73—24no—
154C4×D73—24no—
155C4×D73—88no—
156C4×D73#10214yes—
157C4×D73#10314yes—
158C4×D73#10614yes—
159C4×D73#10714yes—
160#10596833—44no—
161#10596833—44no—
162C22×C73C2×#9511yescharacteristic
163C22×C73C2×#10011yescharacteristic
164C22×C73C2×#10111yescharacteristic
165C22×C73C2×#10211yescharacteristic
166C22×C73C2×#10311yescharacteristic
167C22×C73#41811yescharacteristic
168C22×C73#41911yescharacteristic
169C2×D73—22no—
170C2×D73—22no—
171C2×D73—22no—
172C2×D73—22no—
173C2×D73—22no—
174C2×D73—22no—
175C2×D73#34014yes—
176C4×C73—22no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—82no—
180C4×C73#34014yes—
181Dic73—82no—
182Dic73#34314yes—
183Dic73C2×#10211yescharacteristic
184Dic73C2×#10311yescharacteristic
185Dic73#41712yes—
186Dic73C2×#10611yescharacteristic
187Dic73C2×#10711yescharacteristic
188Dic73#42812yes—
189#8043—731nomaximal, Sylow
190C2×C73#154511yescharacteristic
191C2×C73#154611yescharacteristic
192C2×C73#154811yescharacteristic
193C2×C73#154911yescharacteristic
194C2×C73C2×#34011yescharacteristic
195C2×C73#173611yescharacteristic
196C2×C73C2×#34311yescharacteristic
197D73—24no—
198C2×#431—731no—
199C2×#423—731no—
200C2×#384—731no—
201#1296—734no—
202C73#804311yescharacteristic, complemented, Sylow
203C22×M16—1461no—
204C22×C4⋊C4—732no—
205C22×C4⋊C4—1461no—
206C22×C4⋊C4—1461no—
207C22×C42—731no—
208#431—734no—
209#431—1462no—
210#423—734no—
211#423—1462no—
212C2×C4⋊C8—732no—
213C2×C4⋊C8—732no—
214C2×C4⋊C8—1461no—
215C2×C22⋊C8—1461no—
216C2×C4×C8—1461no—
217#384—732no—
218#384—731no—
219#384—1461no—
220#384—1461no—
221#384#6440811yescharacteristic
222C23×C4—731no—
223C23×C4—731no—
224C23×C4—731no—
225C23×C4—1462no—
226C2×M16—2922no—
227C2×M16—2922no—
228C2×M16—1464no—
229C22×C8—1462no—
230C22×C8—1462no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—2922no—
234C2×C4⋊C4—2922no—
235C2×C4⋊C4—2924no—
236C2×C4⋊C4—2924no—
237C2×C4⋊C4—732no—
238C2×C4⋊C4—1462no—
239C2×C4⋊C4—1462no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1461no—
242C2×C4⋊C4—1461no—
243C2×C4⋊C4C2×#6440812yes—
244C2×C42—731no—
245C2×C42—731no—
246C2×C42—731no—
247C2×C42—732no—
248C2×C42—732no—
249C2×C42—732no—
250C2×C42—732no—
251C2×C42C2×#6440811yescharacteristic
252C4⋊C8—734no—
253C4⋊C8—734no—
254C4⋊C8—1462no—
255C4⋊C8—1462no—
256C4⋊C8—1464no—
257C22⋊C8—1464no—
258C4×C8—1464no—
259C24—731no—
260C22×C4—22no—
261C22×C4—2924no—
262C22×C4—2924no—
263C22×C4—2924no—
264C22×C4—731no—
265C22×C4—731no—
266C22×C4—731no—
267C22×C4—732no—
268C22×C4—732no—
269C22×C4—732no—
270C22×C4—732no—
271C22×C4—732no—
272C22×C4—732no—
273C22×C4—732no—
274C22×C4—732no—
275C22×C4—732no—
276C22×C4—732no—
277C22×C4—1462no—
278C22×C4—1462no—
279C22×C4—1462no—
280C22×C4—1462no—
281C22×C4—1462no—
282C22×C4C22⋊#6440811yescharacteristic
283C22×C4C22⋊#6440811yescharacteristic
284C22×C4C22×#6440811yescharacteristic
285M16—2928no—
286C2×C8—2922no—
287C2×C8—2922no—
288C2×C8—2922no—
289C2×C8—2922no—
290C2×C8—1464no—
291C2×C8—1464no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—2922no—
296C4⋊C4—2922no—
297C4⋊C4—2922no—
298C4⋊C4—2922no—
299C4⋊C4—2924no—
300C4⋊C4—2924no—
301C4⋊C4—2922no—
302C42—732no—
303C42—732no—
304C42—732no—
305C42—732no—
306C42—731no—
307C42—731no—
308C42—731no—
309C42—731no—
310C42C4×#6440812yes—
311C42C4⋊#6440812yes—
312C23—732no—
313C23—732no—
314C23—732no—
315C23—1461no—
316C23—1461no—
317C23—1461no—
318C23—1461no—
319C23C2×C22⋊#6440811yescharacteristic
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—2922no—
328C2×C4—2922no—
329C2×C4—2922no—
330C2×C4—5848no—
331C2×C4—731no—
332C2×C4—731no—
333C2×C4—732no—
334C2×C4—731no—
335C2×C4—731no—
336C2×C4—732no—
337C2×C4—732no—
338C2×C4—731no—
339C2×C4—731no—
340C2×C4—731no—
341C2×C4—731no—
342C2×C4—732no—
343C2×C4—732no—
344C2×C4—734no—
345C2×C4—734no—
346C2×C4—734no—
347C2×C4—734no—
348C2×C4—1464no—
349C2×C4—1464no—
350C2×C4—1464no—
351C2×C4—1464no—
352C2×C4(C2×#64408)⋊C412yes—
353C2×C4(C2×#64408)⋊C412yes—
354C2×C4#1455665312yes—
355C2×C4#1455665412yes—
356C2×C4C2×C4×#6440811yescharacteristic
357C2×C4C2×C4⋊#6440811yescharacteristic
358C8—2924no—
359C8—2924no—
360C22—734no—
361C22—1462no—
362C22—1462no—
363C22—1462no—
364C22—1462no—
365C22—1462no—
366C22—1462no—
367C22C2×(C2×#64408)⋊C411yescharacteristic
368C22D73⋊#10011yescharacteristic
369C22D73⋊#10111yescharacteristic
370C22C2×#1455665311yescharacteristic
371C22C2×#1455665411yescharacteristic
372C22#1798957911yescharacteristic
373C22#1798958011yescharacteristic
374C4—22no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—5842no—
379C4—731no—
380C4—731no—
381C4—731no—
382C4—731no—
383C4—732no—
384C4—732no—
385C4—734no—
386C4#1798946014yes—
387C2—1464no—
388C2#2262543911yescharacteristic
389C2#2262544111yescharacteristic
390C2#2262544511yescharacteristic
391C2#2262544711yescharacteristic
392C2C2×#1798946011yescharacteristic
393C2#2262561811yescharacteristic
394C2#2262563411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#8043
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #22624809C2semidirect
7. #17989605C4semidirect
9. #17989592C4semidirect
32. #14556658C2×C4semidirect
33. #14556658C2×C4semidirect
202. C73#8043semidirect

Maximal subgroups

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

subgroupindex
C2×#179896052
C2×#179895922
#226248092
#384×D732
#804373