Finite Group Databasebeta

← #34003790

Subgroups — #34003790

Order 18688.

528 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625716C211yesmaximal, characteristic
2C4×#14556658C211yesmaximal, characteristic
3C2×#17989447C212yesmaximal
4#22624809C212yesmaximal
5#22612500C211yesmaximal, characteristic
6C2×#14556658C2212yes—
7C2×#14556658C2212yes—
8C2×#14556646—21no—
9C2×C4×#10596833—21no—
10#17989453—21no—
11#17989453—21no—
12#17989447C418yescomplemented
13C2×#14556589C2211yescharacteristic
14C2×Q8×Dic73C411yescharacteristic
15C2×#14556556C411yescharacteristic
16#17988493C411yescharacteristic
17#17988484—21no—
18#17988484—21no—
19#17988456C2211yescharacteristic
20#17988438—21no—
21#17988438—21no—
22C42×Dic73C2211yescharacteristic
23#391×C73C411yescharacteristic
24C22×#10596833—22no—
25C22×#10596833—22no—
26#14556658—22no—
27#14556658—22no—
28#14556658C2×C414yes—
29#14556658C2×C414yes—
30#14556646—42no—
31C4×#10596833—42no—
32C22×Dic146C2×C411yescharacteristic
33#14556589—22no—
34#14556589C2×C414yes—
35Q8×Dic73—42no—
36#14556556—42no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73C2×C411yescharacteristic
39C2×#11924007—21no—
40C2×#11924007—21no—
41C2×#11924007—21no—
42C2×#11924007—21no—
43C2×#11924007—21no—
44C2×C4×Dic73—21no—
45C2×C4×Dic73—21no—
46C2×C4×Dic73C2×C411yescharacteristic
47C2×C4×Dic73D411yescharacteristic
48C2×C4×Dic73D411yescharacteristic
49C2×C4×Dic73D411yescharacteristic
50C2×C4×Dic73D411yescharacteristic
51C2×C4×Dic73C2311yescharacteristic
52#14556498—21no—
53#14556498—21no—
54C22×Q8×C73C2×C411yescharacteristic
55C2×C4⋊C4×C73C2×C411yescharacteristic
56C2×C42×C73C2×C411yescharacteristic
57#95×C73—21no—
58#95×C73—21no—
59C2×#10596833—24no—
60C2×#10596833—24no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C2×#10596833—42no—
65C2×Dic146—21no—
66C2×Dic146—41no—
67C2×Dic146—41no—
68C2×Dic146C4212yes—
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73C22⋊C411yescharacteristic
72C22×Dic73C22⋊C411yescharacteristic
73C22×Dic73C2×D411yescharacteristic
74C22×Dic73C2×D411yescharacteristic
75C4⋊Dic73—42no—
76C4⋊Dic73—41no—
77#11924007—24no—
78#11924007—24no—
79#11924007—42no—
80#11924007—42no—
81#11924007—41no—
82#11924007—41no—
83C4×Dic73—22no—
84C4×Dic73—22no—
85C4×Dic73—21no—
86C4×Dic73—21no—
87C4×Dic73—22no—
88C4×Dic73—42no—
89C4×Dic73—42no—
90C4×Dic73SD1612yes—
91C4×Dic73Q1612yes—
92C4×Dic73C22×C412yes—
93C4×Dic73#4112yes—
94C2×Q8×C73—21no—
95C2×Q8×C73—41no—
96C2×Q8×C73—41no—
97C2×Q8×C73C4212yes—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C4⋊C4×C73—41no—
104C42×C73—41no—
105C23×C73C2×C22⋊C411yescharacteristic
106Q8×C73—42no—
107Q8×C73—42no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—22no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73#10312yes—
122C2×C4×C73C2×C4211yescharacteristic
123C2×C4×C73#11711yescharacteristic
124C2×Dic73—21no—
125C2×Dic73—21no—
126C2×Dic73—21no—
127C2×Dic73—21no—
128C2×Dic73—22no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73#10012yes—
135C2×Dic73#10312yes—
136C2×Dic73C4×D412yes—
137C2×Dic73C4×D412yes—
138C2×Dic73C2×SD1611yescharacteristic
139C2×Dic73C2×Q1611yescharacteristic
140C2×Dic73#13512yes—
141C2×Dic73AGL(1,ℤ/8)11yescharacteristic
142C2×Dic73#13712yes—
143C2×Dic73#13711yescharacteristic
144Dic146—42no—
145Dic146—42no—
146#10596833—44no—
147#10596833—44no—
148C22×C73C4×C22⋊C411yescharacteristic
149C22×C73C2×#10011yescharacteristic
150C22×C73#41311yescharacteristic
151C22×C73C2×#10311yescharacteristic
152C22×C73#41611yescharacteristic
153C22×C73#41711yescharacteristic
154C22×C73#41911yescharacteristic
155C4×C73—22no—
156C4×C73—21no—
157C4×C73—21no—
158C4×C73—41no—
159C4×C73—42no—
160C4×C73—81no—
161Dic73—22no—
162Dic73—22no—
163Dic73—42no—
164Dic73—81no—
165Dic73C4×SD1612yes—
166Dic73C4×Q1612yes—
167Dic73#44012yes—
168Dic73#44112yes—
169#8235—731nomaximal, Sylow
170C2×C73#154811yescharacteristic
171C2×C73#155011yescharacteristic
172C2×C73#155111yescharacteristic
173C2×C73#155411yescharacteristic
174C2×C73C4×#10011yescharacteristic
175C2×C73C4×#10311yescharacteristic
176C2×C73#176311yescharacteristic
177C4×C4⋊Q8—731no—
178#1767—731no—
179C4×C4⋊C8—731no—
180C2×#333—732no—
181#1296—732no—
182C73#823511yescharacteristic, complemented, Sylow
183C2×C4⋊Q8—731no—
184C2×C4×Q8—731no—
185C2×C4×Q8—731no—
186C2×C4⋊C8—732no—
187C2×C4⋊C8—732no—
188C2×#97—1461no—
189C2×C4×C8—1461no—
190#391—731no—
191#391#6440811yescharacteristic
192#389—1461no—
193#389—1461no—
194#384—731no—
195C4×C4⋊C4—1461no—
196C4×C4⋊C4—1461no—
197C43—731no—
198#336—1461no—
199#336—1461no—
200#333—738no—
201C22×Q8—731no—
202C22×Q8C2×#6440811yescharacteristic
203C22×C8—1462no—
204C22×C8—1462no—
205C4⋊Q8—734no—
206C4⋊Q8—1462no—
207C4×Q8—2922no—
208C4×Q8—2922no—
209C2×C4⋊C4—731no—
210C2×C4⋊C4—1461no—
211C2×C4⋊C4—1461no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C4⋊C4—1461no—
216C2×C4⋊C4C2×#6440811yescharacteristic
217C2×C42—731no—
218C2×C42—731no—
219C2×C42—731no—
220C2×C42—731no—
221C2×C42—731no—
222C2×C42—731no—
223C2×C42—1461no—
224C2×C42—1461no—
225C2×C42C2×#6440811yescharacteristic
226C4⋊C8—734no—
227C4⋊C8—734no—
228C4⋊C8—1462no—
229C4⋊C8—1462no—
230#97—2922no—
231C4×C8—2922no—
232#95—21no—
233#95—21no—
234#95—1461no—
235#95—1461no—
236C2×Q8—21no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—2921no—
240C2×Q8—2921no—
241C2×Q8—732no—
242C2×Q8—1461no—
243C2×Q8C4×#6440812yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—731no—
247C22×C4—731no—
248C22×C4—731no—
249C22×C4—731no—
250C22×C4—1461no—
251C22×C4—1461no—
252C22×C4C22⋊#6440811yescharacteristic
253C22×C4C22⋊#6440811yescharacteristic
254C22×C4C22×#6440811yescharacteristic
255C2×C8—2922no—
256C2×C8—2922no—
257C2×C8—2922no—
258C2×C8—2922no—
259C2×C8—1464no—
260C2×C8—1464no—
261C4⋊C4—41no—
262C4⋊C4—2921no—
263C4⋊C4—2922no—
264C4⋊C4—2921no—
265C4⋊C4—2921no—
266C4⋊C4—2922no—
267C4⋊C4—2922no—
268C4⋊C4—1464no—
269C4⋊C4—1464no—
270C42—41no—
271C42—2922no—
272C42—2922no—
273C42—732no—
274C42—732no—
275C42—732no—
276C42—732no—
277C42—1461no—
278C42—1461no—
279C42—1462no—
280C42—1462no—
281C42—1462no—
282C23C2×C22⋊#6440811yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—2922no—
286Q8—2922no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—732no—
306C2×C4—732no—
307C2×C4—731no—
308C2×C4—731no—
309C2×C4—732no—
310C2×C4—731no—
311C2×C4—731no—
312C2×C4—732no—
313C2×C4—732no—
314C2×C4—732no—
315C2×C4—1462no—
316C2×C4—1461no—
317C2×C4—1461no—
318C2×C4—1461no—
319C2×C4—1461no—
320C2×C4#1455665412yes—
321C2×C4C2×C4×#6440811yescharacteristic
322C2×C4#1455667011yescharacteristic
323C8—2924no—
324C8—2924no—
325C22C4×C22⋊#6440811yescharacteristic
326C22D73⋊#10011yescharacteristic
327C22#1798957411yescharacteristic
328C22C2×#1455665411yescharacteristic
329C22#1798957711yescharacteristic
330C22#1798957811yescharacteristic
331C22#1798958011yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—42no—
336C4—41no—
337C4—81no—
338C4—2922no—
339C4—5841no—
340C4—732no—
341C4—732no—
342C4—732no—
343C4—732no—
344C4—1462no—
345C4—1462no—
346C2#2262544511yescharacteristic
347C2#2262544911yescharacteristic
348C2#2262545111yescharacteristic
349C2#2262545711yescharacteristic
350C2#2262568111yescharacteristic
351C2C4×#1455665411yescharacteristic
352C2#2262570311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#391#64408
hypocenterC73#8235
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
12. #17989447C4semidirect
182. C73#8235semidirect

Maximal subgroups

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

subgroupindex
#226257162
C4×#145566582
C2×#179894472
#226248092
#226125002
#823573