Finite Group Databasebeta

← #32980296

Subgroups — #32980296

Order 18688.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618499C211yesmaximal, characteristic
2#22608703C211yesmaximal, characteristic, complemented
3#22608686C211yesmaximal, characteristic, complemented
4#22608313C211yesmaximal, characteristic
5#22607834C211yesmaximal, characteristic, complemented
6#22607772C211yesmaximal, characteristic, complemented
7C73×#1806C211yesmaximal, characteristic
8C2×#14556598—21no—
9#17988640C2211yescharacteristic
10#17988614—22no—
11#17988614—21no—
12#17988611—21no—
13#17988611C2211yescharacteristic
14C2×C4⋊#10596831—21no—
15C2×C4⋊#10596831C2211yescharacteristic
16C2×C4×#10596831C2211yescharacteristic
17#17988337—22no—
18#17988337—21no—
19#17988337—21no—
20Q8⋊#10596831C411yescharacteristic, complemented
21Q8⋊#10596831C411yescharacteristic, complemented
22D4⋊#10596831C411yescharacteristic, complemented
23D4⋊#10596831C411yescharacteristic, complemented
24C4×#41×C73C2211yescharacteristic
25#423×C73C2211yescharacteristic
26C2×#104×C73—21no—
27C4×C4⋊C4×C73C2211yescharacteristic
28#339×C73—21no—
29#14556598—42no—
30#14556598—42no—
31C2×#11924012—22no—
32C2×#11924012—21no—
33C22×#10596831—22no—
34C22×#10596831—21no—
35C22×#10596831—21no—
36C4⋊#10596831—22no—
37C4⋊#10596831—22no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831C2×C411yescharacteristic
41C4⋊#10596831C2×C411yescharacteristic
42C4⋊#10596831D411yescharacteristic
43C4⋊#10596831D411yescharacteristic
44C4⋊#10596831D412yes—
45C22⋊#10596831—21no—
46C4×#10596831—21no—
47C4×#10596831—21no—
48C4×#10596831C2×C411yescharacteristic
49C4×#10596831C2×C411yescharacteristic
50C2×#41×C73D411yescharacteristic
51C2×M16×C73—21no—
52C4×Q8×C73C2×C411yescharacteristic
53C4×Q8×C73Q811yescharacteristic
54C4×D4×C73—21no—
55C4×D4×C73—21no—
56C4×D4×C73C2×C411yescharacteristic
57C4×D4×C73Q811yescharacteristic
58#117×C73—21no—
59#117×C73D411yescharacteristic
60C2×C4⋊C4×C73—21no—
61C2×C42×C73—21no—
62C2×C42×C73—21no—
63C2×C42×C73C2311yescharacteristic
64C4⋊C8×C73—21no—
65C4⋊C8×C73—21no—
66#104×C73—42no—
67#95×C73—21no—
68#11924012—44no—
69#11924012—41no—
70#11924012—41no—
71C2×#10596831—21no—
72C2×#10596831—21no—
73C2×#10596831—21no—
74C2×#10596831—21no—
75C2×#10596831—21no—
76C2×#10596831—22no—
77C2×#10596831—21no—
78C2×#10596831—42no—
79C2×#10596831—44no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82#41×C73—21no—
83#41×C73—21no—
84#41×C73SD1612yes—
85#41×C73SD1612yes—
86C2×Q8×C73C4⋊C411yescharacteristic
87C2×D4×C73—21no—
88C2×D4×C73C4⋊C411yescharacteristic
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—41no—
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73#4111yescharacteristic
95C22×C4×C73#4111yescharacteristic
96M16×C73—41no—
97M16×C73—41no—
98C2×C8×C73—41no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73—21no—
101C4⋊C4×C73—22no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73C4⋊C411yescharacteristic
104C4⋊C4×C73C4⋊C411yescharacteristic
105C22⋊C4×C73—21no—
106C22⋊C4×C73—21no—
107C22⋊C4×C73—21no—
108C42×C73—21no—
109C42×C73—21no—
110C42×C73—21no—
111C42×C73—41no—
112C42×C73—41no—
113C42×C73C22×C411yescharacteristic
114C42×C73C2×D411yescharacteristic
115C42×C73C2×D411yescharacteristic
116C42×C73C2×Q811yescharacteristic
117C23×C73—21no—
118C23×C73#12311yescharacteristic
119Q8×C73—21no—
120Q8×C73#10612yes—
121D4×C73—22no—
122D4×C73—22no—
123D4×C73—21no—
124D4×C73#10612yes—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—22no—
129C2×C4×C73—22no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—42no—
136C2×C4×C73—81no—
137C2×C4×C73—81no—
138C2×C4×C73—81no—
139C2×C4×C73C2×C4⋊C411yescharacteristic
140C2×C4×C73C4×D411yescharacteristic
141C2×C4×C73C4×D411yescharacteristic
142C2×C4×C73#12011yescharacteristic
143C2×C4×C73C22⋊Q811yescharacteristic
144C2×C4×C73C22⋊Q811yescharacteristic
145C2×C4×C73C2×SD1611yescharacteristic
146C2×C4×C73C2×SD1611yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73#13711yescharacteristic
149C8×C73—81no—
150#10596831—22no—
151#10596831—41no—
152#10596831—41no—
153#10596831—41no—
154#10596831—41no—
155#10596831—42no—
156#10596831—41no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—22no—
160C22×C73—41no—
161C22×C73—41no—
162C22×C73#38511yescharacteristic
163C22×C73#48111yescharacteristic
164C22×C73#48311yescharacteristic
165C4×C73—22no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—41no—
170C4×C73—41no—
171C4×C73—82no—
172C4×C73C2×#10611yescharacteristic
173C4×C73#42811yescharacteristic
174C4×C73#44311yescharacteristic
175C4×C73#44411yescharacteristic
176C4×C73#44811yescharacteristic
177C4×C73#45011yescharacteristic
178#8904—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—42no—
182C2×C73#185311yescharacteristic
183C2×C73#186111yescharacteristic
184C2×C73#186611yescharacteristic
185#2906—731no—
186#1806D7311yescharacteristic
187#1570—731no—
188#1566—731no—
189#1476—731no—
190#1303—731no—
191#1279—731no—
192C73#890411yescharacteristic, complemented, Sylow
193C2×#131—1461no—
194C4×#41C2×D7311yescharacteristic
195#431—731no—
196#424—1462no—
197#424—1461no—
198#423—731no—
199#423—1461no—
200#423C2×D7311yescharacteristic
201C2×C4⋊C8—731no—
202C2×C4⋊C8—1461no—
203C2×#104—21no—
204C2×C4×C8—731no—
205C4×C4⋊C4C2×D7311yescharacteristic
206#339—21no—
207#334—1462no—
208#334—1461no—
209#334—1461no—
210Q8⋊C8—731no—
211Q8⋊C8—731no—
212D4⋊C8—731no—
213D4⋊C8—731no—
214C2×#41C22⋊D7311yescharacteristic
215#131—2922no—
216#131—2922no—
217C2×M16—21no—
218C2×M16—1462no—
219C2×M16—1461no—
220C22×C8—1462no—
221C22×C8—1461no—
222C22×C8—1461no—
223C4×Q8Dic14611yescharacteristic
224C4×Q8C4×D7311yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic14611yescharacteristic
228C4×D4C4×D7311yescharacteristic
229#117—21no—
230#117C22⋊D7311yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×D7311yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—731no—
238C4⋊C8—731no—
239C4⋊C8—731no—
240C4⋊C8—731no—
241C4⋊C8—732no—
242C4⋊C8—1462no—
243C4⋊C8—1462no—
244C4⋊C8—1461no—
245C4⋊C8—1461no—
246#104—42no—
247C22⋊C8—1461no—
248C4×C8—731no—
249C4×C8—731no—
250C4×C8—1461no—
251C4×C8—1461no—
252#95—21no—
253#41—21no—
254#41—21no—
255#41#1192401512yes—
256#41#1192401612yes—
257C2×Q8#1192400711yescharacteristic
258C2×D4—21no—
259C2×D4#1192400711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4C2×C22⋊D7311yescharacteristic
265C22×C4#1192402711yescharacteristic
266C22×C4#1192402711yescharacteristic
267M16—41no—
268M16—41no—
269M16—2924no—
270M16—2921no—
271M16—2921no—
272C2×C8—41no—
273C2×C8—2924no—
274C2×C8—2921no—
275C2×C8—2921no—
276C2×C8—2922no—
277C2×C8—1461no—
278C2×C8—1461no—
279C2×C8—1461no—
280C2×C8—1461no—
281C2×C8—1461no—
282C2×C8—1462no—
283C2×C8—1461no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4#1192400711yescharacteristic
289C4⋊C4#1192400711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C2×C4×D7311yescharacteristic
299C42D4×D7311yescharacteristic
300C42D4×D7311yescharacteristic
301C42C2×Dic14611yescharacteristic
302C23—21no—
303C23#1455657211yescharacteristic
304Q8—21no—
305Q8#1455652212yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1455652212yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×#1192400711yescharacteristic
325C2×C4#1455655111yescharacteristic
326C2×C4#1455655111yescharacteristic
327C2×C4#1455655911yescharacteristic
328C2×C4#1455656811yescharacteristic
329C2×C4#1455656811yescharacteristic
330C2×C4C2×#1192401511yescharacteristic
331C2×C4C2×#1192401611yescharacteristic
332C2×C4#1455661611yescharacteristic
333C2×C4#1455662111yescharacteristic
334C8—81no—
335C8—2921no—
336C8—2921no—
337C8—2921no—
338C8—2921no—
339C8—2922no—
340C8—2921no—
341C8—1462no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1798845911yescharacteristic
348C22#1798890311yescharacteristic
349C22#1798891511yescharacteristic
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×#1455652211yescharacteristic
358C4#1798863011yescharacteristic
359C4#1798870111yescharacteristic
360C4#1798870611yescharacteristic
361C4#1798872511yescharacteristic
362C4#1798873711yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2261012311yescharacteristic
367C2#2261016911yescharacteristic
368C2#2261020411yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1806C2
socleC22×C73#385
prime radicalC2×#41×C73D4
cosocleC2×C42×C73C23
hypercenter#1806D73
hypocenterC73#8904
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #22608703C2semidirect
3. #22608686C2semidirect
5. #22607834C2semidirect
6. #22607772C2semidirect
20. Q8⋊#10596831C4semidirect
21. Q8⋊#10596831C4semidirect
22. D4⋊#10596831C4semidirect
23. D4⋊#10596831C4semidirect
192. C73#8904semidirect

Maximal subgroups

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

subgroupindex
#226184992
#226087032
#226086862
#226083132
#226078342
#226077722
C73×#18062
#890473