Finite Group Databasebeta

← #33057409

Subgroups — #33057409

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618939C212yesmaximal, complemented
2C4×#14556610C211yesmaximal, characteristic
3#22608829C211yesmaximal, characteristic, complemented
4#22608827C211yesmaximal, characteristic, complemented
5#22608767C212yesmaximal, complemented
6#22608764C212yesmaximal, complemented
7D8×#10596831C211yesmaximal, characteristic, complemented
8#22608693C211yesmaximal, characteristic, complemented
9#22608362C212yesmaximal, complemented
10C8×#11924011C211yesmaximal, characteristic, complemented
11C8×D8×C73C211yesmaximal, characteristic, complemented
12C2×#14556610C411yescharacteristic
13C4×#11924027C2212yes—
14#17988717C2212yes—
15#17988714C2212yes—
16C4×Dic292C2211yescharacteristic
17#17988681C411yescharacteristic
18#17988678C412yes—
19C4×#11924016C2212yes—
20D8×Dic73C411yescharacteristic
21#17988658—22no—
22#17988655—22no—
23D4×#10596831C2212yes—
24#17988643C2212yes—
25#17988641—22no—
26#17988638C2211yescharacteristic
27#17988624C411yescharacteristic
28#17988580C412yes—
29C4×#11924011C2211yescharacteristic
30C2×#14556501—22no—
31C4×C8×D73C2211yescharacteristic
32#17988340C2211yescharacteristic
33#17988337C2212yes—
34#17988319C2212yes—
35#17988317C2212yes—
36D4⋊#10596831C2212yes—
37#17988309C2211yescharacteristic
38#17988308C2211yescharacteristic
39C8×#10596831C2211yescharacteristic
40C4×D8×C73C2211yescharacteristic
41C8×D4×C73C2212yes—
42#335×C73C2211yescharacteristic
43D4⋊C8×C73C2212yes—
44C82×C73C2211yescharacteristic
45C2×#11924027C2×C412yes—
46#14556610—24no—
47C2×Dic292C2×C411yescharacteristic
48C2×#11924016C2×C412yes—
49C2×#11924011D411yescharacteristic
50C2×#11924011D411yescharacteristic
51C22×#10596831—22no—
52C2×C8×D73C2×C411yescharacteristic
53C2×C8×D73C2×C411yescharacteristic
54C4×Dic146C2312yes—
55#14556556C2×C412yes—
56D4×Dic73C2×C412yes—
57C4×C22⋊D73—22no—
58#14556551—22no—
59#14556549C2×C412yes—
60#14556547—22no—
61C2×C4×Dic73—22no—
62C42×D73C2311yescharacteristic
63#14556525C2×C411yescharacteristic
64#14556522C2×C412yes—
65C4⋊#10596831C2312yes—
66#14556520—22no—
67#14556520—21no—
68#14556520—22no—
69#14556516C2×C412yes—
70#14556514C2×C412yes—
71#14556512C2×C412yes—
72C22⋊#10596831—22no—
73#14556503—22no—
74#14556503—21no—
75#14556501—22no—
76#14556501—22no—
77#14556501—22no—
78#14556501D411yescharacteristic
79#14556501D411yescharacteristic
80C4×#10596831C2311yescharacteristic
81C4×#10596831C2311yescharacteristic
82C4×#10596831C2311yescharacteristic
83C8×Dic73C2×C411yescharacteristic
84C8×Dic73C2×C411yescharacteristic
85C2×D8×C73C2×C411yescharacteristic
86C22×C8×C73—22no—
87C4×D4×C73C2312yes—
88#107×C73C2×C411yescharacteristic
89C4⋊C8×C73C2312yes—
90#102×C73C2×C412yes—
91C22⋊C8×C73—22no—
92C4×C8×C73C2311yescharacteristic
93C4×C8×C73C2311yescharacteristic
94C4×C8×C73C2311yescharacteristic
95#11924027—24no—
96#11924027—42no—
97C2×Dic146C22×C412yes—
98C2×C22⋊D73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C2×C4×D73C22×C411yescharacteristic
102C2×C4×D73C2×D411yescharacteristic
103C2×C4×D73#4111yescharacteristic
104Dic292—22no—
105#11924016—24no—
106#11924011—22no—
107#11924011—22no—
108C2×#10596831—22no—
109C2×#10596831—21no—
110C2×#10596831—44no—
111C2×#10596831C22×C411yescharacteristic
112C2×#10596831C22×C411yescharacteristic
113C2×#10596831C2×D411yescharacteristic
114C2×#10596831C2×D411yescharacteristic
115C8×D73—22no—
116C8×D73—41no—
117C4⋊Dic73C22×C412yes—
118#11924007—22no—
119#11924007—22no—
120C22⋊Dic73—22no—
121#11924005—22no—
122C4×Dic73—22no—
123C4×Dic73—22no—
124C4×Dic73—22no—
125C4×Dic73C22×C411yescharacteristic
126C4×Dic73C2×D411yescharacteristic
127C4×Dic73#4111yescharacteristic
128C2×D4×C73C22×C412yes—
129C22×C4×C73—22no—
130D8×C73M1614yescomplemented
131C2×C8×C73—22no—
132C2×C8×C73—21no—
133C2×C8×C73—44no—
134C2×C8×C73C22×C411yescharacteristic
135C2×C8×C73C22×C411yescharacteristic
136C2×C8×C73C2×D411yescharacteristic
137C2×C8×C73C2×D411yescharacteristic
138C4⋊C4×C73C22×C412yes—
139C22⋊C4×C73—22no—
140C42×C73C2411yescharacteristic
141C23×C73—22no—
142C22×D73C4×D411yescharacteristic
143D4×C73—22no—
144D4×C73C2×M1614yes—
145C2×C4×C73—22no—
146C2×C4×C73—42no—
147C2×C4×C73—42no—
148C2×C4×C73C23×C411yescharacteristic
149C2×C4×C73C22×D411yescharacteristic
150C2×C4×C73C2×#4111yescharacteristic
151C22⋊D73—44no—
152C2×Dic73—22no—
153C2×Dic73—22no—
154C2×Dic73—44no—
155C2×Dic73—82no—
156C2×Dic73C4×D411yescharacteristic
157C2×Dic73C4×D411yescharacteristic
158C2×Dic73C4×D411yescharacteristic
159C4×D73—21no—
160C4×D73—21no—
161C4×D73—21no—
162C4×D73—21no—
163C4×D73—41no—
164Dic146—22no—
165Dic146—22no—
166C8×C73—21no—
167C8×C73—21no—
168C8×C73—41no—
169C8×C73—42no—
170C8×C73C2×M1612yes—
171C8×C73#13512yescomplemented
172#10596831—21no—
173#10596831—21no—
174#10596831—21no—
175#10596831—41no—
176#10596831—42no—
177#10596831#13512yescomplemented
178C22×C73—24no—
179C22×C73—42no—
180C22×C73—42no—
181C22×C73C2×C4×D411yescharacteristic
182C2×D73—21no—
183C2×D73—21no—
184C2×D73—21no—
185C4×C73—21no—
186C4×C73—42no—
187C4×C73C22×M1611yescharacteristic
188C4×C73#56811yescharacteristic
189C4×C73C2×#13511yescharacteristic
190C4×C73C2×#13511yescharacteristic
191Dic73—21no—
192Dic73—21no—
193Dic73—21no—
194Dic73—42no—
195Dic73—42no—
196#18035—731nomaximal, Sylow
197C2×C73—44no—
198C2×C73D4×M1611yescharacteristic
199C2×C73C4×#13511yescharacteristic
200C2×C73#295711yescharacteristic
201D73—41no—
202#2970—732no—
203C4×#135—731no—
204#1591—731no—
205#1591—731no—
206#1582—732no—
207#1582—732no—
208C8×D8—731no—
209C8×D8D7311yescharacteristic, complemented
210#1568—731no—
211#1488—732no—
212C8×M16—731no—
213C73#1803511yescharacteristic, complemented, Sylow
214C2×#135—731no—
215C4×#41—732no—
216#446—732no—
217#446—732no—
218C4×Q16—731no—
219C4×Q16—731no—
220C4×SD16—732no—
221C4×SD16—732no—
222C4×D8—731no—
223C4×D8C2×D7311yescharacteristic
224#435—1462no—
225#435—1462no—
226C8×D4—732no—
227C8×D4C2×D7312yes—
228#432—732no—
229#432—1462no—
230#431—731no—
231#427—731no—
232#416—732no—
233C4×M16—731no—
234C2×#97—1462no—
235C2×C4×C8—731no—
236#335—731no—
237#335C2×D7311yescharacteristic
238#334—732no—
239Q8⋊C8—732no—
240Q8⋊C8—732no—
241D4⋊C8—732no—
242D4⋊C8C2×D7312yes—
243#322—731no—
244#322—731no—
245C82—731no—
246C82C2×D7311yescharacteristic
247C2×#41—732no—
248#135—1464no—
249C2×Q16—731no—
250C2×SD16—732no—
251C2×D8C4×D7311yescharacteristic
252C2×M16—731no—
253C2×M16—731no—
254C22×C8—22no—
255C22×C8—731no—
256C22×C8—731no—
257C22×C8—1462no—
258C4×Q8—732no—
259C4×Q8—732no—
260C4×D4—732no—
261C4×D4—1462no—
262C4×D4—1462no—
263C4×D4C22×D7312yes—
264#117—732no—
265#117—1462no—
266C2×C42—731no—
267C2×C42—1462no—
268#107—731no—
269#107C4×D7311yescharacteristic
270#106—732no—
271C4⋊C8—732no—
272C4⋊C8—1461no—
273C4⋊C8—1462no—
274C4⋊C8—1462no—
275C4⋊C8C22×D7312yes—
276#103—732no—
277#103—732no—
278#102—732no—
279#102C4×D7312yes—
280C22⋊C8—22no—
281C22⋊C8—1461no—
282C22⋊C8—1462no—
283C22⋊C8—1462no—
284#97—731no—
285#97—731no—
286#97—1462no—
287#97—1462no—
288#97—1462no—
289C4×C8—731no—
290C4×C8—731no—
291C4×C8—731no—
292C4×C8—731no—
293C4×C8—731no—
294C4×C8C22×D7311yescharacteristic
295C4×C8C22×D7311yescharacteristic
296C4×C8C22×D7311yescharacteristic
297#41—2922no—
298#41—1464no—
299C2×Q8—732no—
300C2×D4—1462no—
301C2×D4C2×C4×D7312yes—
302C22×C4—22no—
303C22×C4—731no—
304C22×C4—731no—
305C22×C4—731no—
306C22×C4—1462no—
307C22×C4—1462no—
308Q16—1462no—
309SD16—1464no—
310D8#1192401114yescomplemented
311M16—1462no—
312M16—1462no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—44no—
316C2×C8—2921no—
317C2×C8—2924no—
318C2×C8—731no—
319C2×C8—731no—
320C2×C8—731no—
321C2×C8—731no—
322C2×C8—1462no—
323C2×C8—1461no—
324C2×C8—1462no—
325C2×C8C2×C4×D7311yescharacteristic
326C2×C8C2×C4×D7311yescharacteristic
327C2×C8D4×D7311yescharacteristic
328C2×C8D4×D7311yescharacteristic
329C4⋊C4—732no—
330C4⋊C4—1462no—
331C4⋊C4—1462no—
332C4⋊C4C2×C4×D7312yes—
333C22⋊C4—22no—
334C22⋊C4—1462no—
335C22⋊C4—1462no—
336C42—731no—
337C42—731no—
338C42—731no—
339C42—1462no—
340C42—1462no—
341C42—1462no—
342C42C23×D7311yescharacteristic
343C23—22no—
344C23—731no—
345Q8—1462no—
346Q8—1462no—
347D4—22no—
348D4—2924no—
349D4C2×#1192401114yes—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—2921no—
354C2×C4—2924no—
355C2×C4—5842no—
356C2×C4—731no—
357C2×C4—731no—
358C2×C4—731no—
359C2×C4—1462no—
360C2×C4—1462no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4—1461no—
364C2×C4—1461no—
365C2×C4C22×C4×D7311yescharacteristic
366C2×C4C2×D4×D7311yescharacteristic
367C2×C4#41×D7311yescharacteristic
368C8—21no—
369C8—21no—
370C8—41no—
371C8—42no—
372C8—2921no—
373C8—2922no—
374C8—732no—
375C8—1461no—
376C8—1461no—
377C8—1461no—
378C8C2×#1192401112yes—
379C8#1455661012yescomplemented
380C22—24no—
381C22—42no—
382C22—42no—
383C22—1461no—
384C22—1461no—
385C22—1461no—
386C22C4×D4×D7311yescharacteristic
387C4—21no—
388C4—42no—
389C4—2922no—
390C4—2922no—
391C4—1461no—
392C4—1461no—
393C4—1461no—
394C4C22×#1192401111yescharacteristic
395C4#1798935211yescharacteristic
396C4#135×D7311yescharacteristic
397C4C2×#1455661011yescharacteristic
398C2—44no—
399C2—2921no—
400C2D4×#1192401111yescharacteristic
401C2C4×#1455661011yescharacteristic
402C2#2261884311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×D8×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556610C4
cosocleC42×C73C24
hypercenterC8×D8D73
hypocenterC73#18035
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
1. #22618939C2semidirect
3. #22608829C2semidirect
4. #22608827C2semidirect
5. #22608767C2semidirect
6. #22608764C2semidirect
7. D8×#10596831C2semidirect
8. #22608693C2semidirect
9. #22608362C2semidirect
10. C8×#11924011C2semidirect
11. C8×D8×C73C2semidirect
130. D8×C73M16semidirect
171. C8×C73#135semidirect
177. #10596831#135semidirect
209. C8×D8D73semidirect
213. C73#18035semidirect
310. D8#11924011semidirect
379. C8#14556610semidirect

Maximal subgroups

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

subgroupindex
#226189392
C4×#145566102
#226088292
#226088272
#226087672
#226087642
D8×#105968312
#226086932
#226083622
C8×#119240112
C8×D8×C732
#1803573