Finite Group Databasebeta

← #34018210

Subgroups — #34018210

Order 18688.

612 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626862C212yesmaximal
2#22625509C212yesmaximal, complemented
3#22625507C214yesmaximal, complemented
4#22625490C211yesmaximal, characteristic, complemented
5#22625488C212yesmaximal, complemented
6#22625480C212yesmaximal, complemented
7#22625479C211yesmaximal, characteristic, complemented
8C8×#11924027C211yesmaximal, characteristic
9#17989636—22no—
10#17989622—22no—
11#17989620—24no—
12#17989606C2212yes—
13#17989603C2214yes—
14#17989591C2212yes—
15#17989590C2214yes—
16#17989451C2211yescharacteristic
17#17989451C2211yescharacteristic
18#17989451C2212yes—
19#17989451C2212yes—
20#17989450C2214yes—
21#17989449C2214yes—
22#17989448C2212yes—
23C4×#11924027C2211yescharacteristic
24C8×Dic146C411yescharacteristic
25#17988711C411yescharacteristic
26D4×#10596831C411yescharacteristic
27C8×C22⋊D73C412yes—
28#17988650C412yes—
29#17988639C2211yescharacteristic
30#17988637C2212yes—
31C2×C8×Dic73C2212yes—
32C4×C8×D73C2211yescharacteristic
33C8×D4×C73C411yescharacteristic
34C2×#11924038—22no—
35C2×#11924037—24no—
36#14556658—24no—
37C4⋊#10596833—22no—
38C4⋊#10596833C2312yes—
39C4⋊#10596833C2312yes—
40C4⋊#10596833C2314yes—
41#14556648—22no—
42C22⋊#10596833—24no—
43C4×#10596833—24no—
44C4×#10596833—22no—
45C2×#11924027Q811yescharacteristic
46C22×#10596831C2×C412yes—
47C2×C8×D73C2×C411yescharacteristic
48C2×C8×D73C2×C411yescharacteristic
49C4×Dic146C2×C411yescharacteristic
50#14556556D411yescharacteristic
51D4×Dic73D411yescharacteristic
52C4×C22⋊D73C2×C412yes—
53#14556551D412yes—
54#14556549Q811yescharacteristic
55#14556547Q812yes—
56C2×C4×Dic73C2312yes—
57C42×D73C2311yescharacteristic
58C4⋊#10596831C2×C411yescharacteristic
59#14556520C2×C412yes—
60#14556520C2×C412yes—
61C22⋊#10596831C2×C412yes—
62#14556503C2×C412yes—
63C4×#10596831C2×C411yescharacteristic
64C8×Dic73C2×C412yes—
65C8×Dic73C2×C412yes—
66C8×Dic73C2311yescharacteristic
67C8×Dic73C2312yes—
68C8×Dic73C2311yescharacteristic
69C22×C8×C73C2×C412yes—
70C4×D4×C73C2×C411yescharacteristic
71C4⋊C8×C73C2×C411yescharacteristic
72C22⋊C8×C73C2×C412yes—
73C4×C8×C73C2×C411yescharacteristic
74#11924038—82no—
75#11924037—84no—
76C2×#10596833—22no—
77C2×#10596833—24no—
78C2×#10596833—22no—
79#11924027—24no—
80C2×Dic146C4⋊C411yescharacteristic
81C2×C22⋊D73C4⋊C412yes—
82C22×Dic73C2×D412yes—
83C22×Dic73C2×Q812yes—
84C2×C4×D73C22×C411yescharacteristic
85C2×C4×D73C2×D411yescharacteristic
86C2×C4×D73C2×Q811yescharacteristic
87C2×#10596831—24no—
88C2×#10596831C22×C412yes—
89C2×#10596831C22×C411yescharacteristic
90C2×#10596831C22×C411yescharacteristic
91C8×D73—22no—
92C8×D73—41no—
93C4⋊Dic73C4⋊C411yescharacteristic
94#11924007C4⋊C412yes—
95#11924007C4⋊C412yes—
96C22⋊Dic73C4⋊C412yes—
97#11924005C4⋊C412yes—
98C4×Dic73C22×C412yes—
99C4×Dic73C2×D411yescharacteristic
100C4×Dic73C2×D412yes—
101C4×Dic73C2×Q811yescharacteristic
102C4×Dic73C2×Q812yes—
103C4×Dic73C2411yescharacteristic
104C2×D4×C73C4⋊C411yescharacteristic
105C22×C4×C73C22×C412yes—
106C2×C8×C73—42no—
107C2×C8×C73C22×C411yescharacteristic
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73C22×C412yes—
110C4⋊C4×C73C4⋊C411yescharacteristic
111C22⋊C4×C73C4⋊C412yes—
112C42×C73C22×C411yescharacteristic
113C23×C73C2×C4⋊C412yes—
114C22×D73C2×C4⋊C411yescharacteristic
115D4×C73—22no—
116C2×C4×C73—22no—
117C2×C4×C73—22no—
118C2×C4×C73C2×C4⋊C411yescharacteristic
119C2×C4×C73C2×C4⋊C412yes—
120C2×C4×C73C2×C4⋊C411yescharacteristic
121C2×C4×C73C23×C411yescharacteristic
122C22⋊D73—24no—
123C2×Dic73—22no—
124C2×Dic73—22no—
125C2×Dic73—24no—
126C2×Dic73C2×C4⋊C412yes—
127C2×Dic73C2×C4⋊C411yescharacteristic
128C2×Dic73C2×C4⋊C412yes—
129C2×Dic73C22×D411yescharacteristic
130C2×Dic73C22×Q811yescharacteristic
131C4×D73—21no—
132C4×D73—21no—
133C4×D73—22no—
134C4×D73—21no—
135C4×D73—21no—
136Dic146#10814yes—
137#10596833—42no—
138#10596833—44no—
139#10596833—42no—
140C8×C73—21no—
141C8×C73—21no—
142C8×C73—22no—
143#10596831—22no—
144#10596831—21no—
145#10596831#13112yes—
146C22×C73—22no—
147C22×C73—22no—
148C22×C73—22no—
149C22×C73C22×C4⋊C411yescharacteristic
150C2×D73—21no—
151C2×D73—21no—
152C2×D73—21no—
153C4×C73—22no—
154C4×C73—21no—
155C4×C73C2×#10812yes—
156C4×C73C2×#13111yescharacteristic
157C4×C73#56811yescharacteristic
158Dic73—21no—
159Dic73—22no—
160Dic73—21no—
161Dic73C2×#10814yes—
162Dic73#57611yescharacteristic
163Dic73#57811yescharacteristic
164#17837—731nomaximal, Sylow
165C2×C73—42no—
166C2×C73#290611yescharacteristic
167C2×C73#291311yescharacteristic
168C2×C73C22×#10811yescharacteristic
169D73—41no—
170#2991—732no—
171C8×#41—731no—
172#1574—734no—
173#1574—732no—
174#1567—732no—
175#1567—731no—
176#1564—731no—
177#1564—732no—
178C73#1783711yescharacteristic, complemented, Sylow
179C4×#41—731no—
180C8×Q8—731no—
181C8×Q8—731no—
182C8×Q8—1462no—
183#436—1464no—
184#436—1462no—
185C8×D4—732no—
186C8×D4—731no—
187C8×D4—732no—
188C8×D4#6440811yescharacteristic
189#431—731no—
190#431—732no—
191#431—734no—
192#431—732no—
193#423—734no—
194#423—732no—
195C2×C4×C8—732no—
196C2×C4×C8—731no—
197#335—731no—
198#335—731no—
199#335—732no—
200#335—734no—
201#335—732no—
202#334—732no—
203#334—734no—
204C2×#41—731no—
205C2×M16—1464no—
206C2×M16—1462no—
207C22×C8—731no—
208C22×C8—732no—
209C22×C8—731no—
210C22×C8C2×#6440812yes—
211C4×Q8—731no—
212C4×Q8—731no—
213C4×D4—731no—
214C4×D4—732no—
215C4×D4—732no—
216C4×D4C2×#6440811yescharacteristic
217#117—731no—
218#117—732no—
219C2×C42—732no—
220C2×C42—731no—
221C4⋊C8—732no—
222C4⋊C8—732no—
223C4⋊C8—732no—
224C4⋊C8—732no—
225C4⋊C8—734no—
226C4⋊C8—731no—
227C4⋊C8—1462no—
228C4⋊C8—1464no—
229C4⋊C8C2×#6440811yescharacteristic
230C22⋊C8—732no—
231C22⋊C8—732no—
232C22⋊C8—1464no—
233C22⋊C8—1462no—
234C22⋊C8C2×#6440812yes—
235C4×C8—732no—
236C4×C8—731no—
237C4×C8—731no—
238C4×C8—732no—
239C4×C8—732no—
240C4×C8—731no—
241C4×C8—1464no—
242C4×C8—1462no—
243C4×C8C2×#6440811yescharacteristic
244#41—1464no—
245C2×Q8—731no—
246C2×D4—732no—
247C2×D4C4⋊#6440811yescharacteristic
248C22×C4—732no—
249C22×C4—732no—
250C22×C4—731no—
251C22×C4—731no—
252C22×C4—731no—
253C22×C4C22×#6440812yes—
254M16—5844no—
255M16—5842no—
256C2×C8—42no—
257C2×C8—2921no—
258C2×C8—731no—
259C2×C8—732no—
260C2×C8—731no—
261C2×C8—1462no—
262C2×C8—1462no—
263C2×C8—1462no—
264C2×C8—1464no—
265C2×C8—1464no—
266C2×C8C22×#6440811yescharacteristic
267C2×C8C22×#6440812yes—
268C2×C8C22×#6440811yescharacteristic
269C4⋊C4—732no—
270C4⋊C4—732no—
271C4⋊C4—731no—
272C4⋊C4C4⋊#6440811yescharacteristic
273C22⋊C4—732no—
274C22⋊C4—732no—
275C22⋊C4C4⋊#6440812yes—
276C42—732no—
277C42—731no—
278C42—732no—
279C42—731no—
280C42—732no—
281C42—731no—
282C42C22×#6440811yescharacteristic
283C23—731no—
284C23C2×C4⋊#6440812yes—
285Q8—734no—
286D4—22no—
287D4—1464no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—731no—
291C2×C4—732no—
292C2×C4—731no—
293C2×C4—731no—
294C2×C4—732no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4—1462no—
298C2×C4—1464no—
299C2×C4—1462no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1462no—
303C2×C4C2×C4⋊#6440812yes—
304C2×C4C2×C4⋊#6440811yescharacteristic
305C2×C4C2×C4⋊#6440811yescharacteristic
306C2×C4C23×#6440811yescharacteristic
307C8—21no—
308C8—21no—
309C8—22no—
310C8—2922no—
311C8—2924no—
312C8—2922no—
313C8—732no—
314C8—1462no—
315C8—1461no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—1461no—
320C22—1461no—
321C22—1461no—
322C22C22×C4⋊#6440811yescharacteristic
323C4—22no—
324C4—21no—
325C4—731no—
326C4—731no—
327C4—734no—
328C4—1461no—
329C4—1462no—
330C4—1461no—
331C4D73⋊#10812yes—
332C4D73⋊#13111yescharacteristic
333C4#1798966511yescharacteristic
334C2—42no—
335C2—2921no—
336C2#2262657811yescharacteristic
337C2#2262658611yescharacteristic
338C2C2×D73⋊#10811yescharacteristic
subgroupG/H
centerC22C22×C4⋊#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
FittingC8×D4×C73C4
socleC22×C73C22×C4⋊C4
prime radicalC2×#11924027Q8
cosocleC4×Dic73C24
hypercenterC8×D4#64408
hypocenterC73#17837
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
2. #22625509C2semidirect
3. #22625507C2semidirect
4. #22625490C2semidirect
5. #22625488C2semidirect
6. #22625480C2semidirect
7. #22625479C2semidirect
178. C73#17837semidirect

Maximal subgroups

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

subgroupindex
#226268622
#226255092
#226255072
#226254902
#226254882
#226254802
#226254792
C8×#119240272
#1783773