Finite Group Databasebeta

← #33004686

Subgroups — #33004686

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556535C211yesmaximal, characteristic, complemented
2C2×#17988828C211yesmaximal, characteristic, complemented
3C2×#17988390C211yesmaximal, characteristic, complemented
4C2×#17988388C211yesmaximal, characteristic, complemented
5C2×C22⋊#11924003C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C2×#466×C73C211yesmaximal, characteristic
8C22×Dic292C2211yescharacteristic
9C2×#14556570—21no—
10C2×#14556535—22no—
11C2×#14556535—22no—
12C2×#14556535—22no—
13C22×#11924003C2211yescharacteristic
14C22×#11924003C2211yescharacteristic
15#17988828—22no—
16#17988828—22no—
17C2×#14556525C2211yescharacteristic
18C2×#14556516—21no—
19#17988390—22no—
20#17988388—22no—
21C22⋊#11924003—22no—
22C22×D8×C73C2211yescharacteristic
23C23×C8×C73C2211yescharacteristic
24C2×#121×C73—21no—
25#466×C73—22no—
26#466×C73—22no—
27C2×#107×C73C2211yescharacteristic
28C2×#102×C73—21no—
29C22×Dic146—21no—
30C2×Dic292—21no—
31C2×Dic292—21no—
32C2×Dic292—21no—
33C2×Dic292—21no—
34C2×Dic292—21no—
35C2×Dic292D412yescomplemented
36#14556570—24no—
37#14556570—42no—
38C2×C4⋊Dic73—21no—
39C2×#11924007—41no—
40C2×C22⋊Dic73—41no—
41#14556535—28no—
42C2×#11924003—21no—
43C2×#11924003—21no—
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47C2×#11924003D412yescomplemented
48#14556525—21no—
49#14556525D412yescomplemented
50#14556516—24no—
51C22×D4×C73—21no—
52C22×D4×C73—41no—
53C23×C4×C73D411yescharacteristic
54C2×D8×C73—21no—
55C2×D8×C73—21no—
56C2×D8×C73—21no—
57C2×D8×C73—21no—
58C2×D8×C73—21no—
59C2×D8×C73D412yes—
60C22×C8×C73—21no—
61C22×C8×C73—21no—
62C22×C8×C73—21no—
63C22×C8×C73—21no—
64C22×C8×C73—21no—
65C22×C8×C73—21no—
66C22×C8×C73D411yescharacteristic
67C22×C8×C73C2311yescharacteristic
68#121×C73—24no—
69#121×C73—42no—
70C2×C4⋊C4×C73—21no—
71C2×C22⋊C4×C73—41no—
72#107×C73—21no—
73#107×C73D412yes—
74#102×C73—24no—
75C2×Dic146—22no—
76C2×Dic146—41no—
77C2×Dic146—41no—
78C2×Dic146—41no—
79C2×Dic146—41no—
80C2×Dic146—41no—
81C22×Dic73—41no—
82C22×Dic73—41no—
83Dic292—22no—
84Dic292—22no—
85Dic292—41no—
86Dic292—41no—
87C4⋊Dic73—22no—
88C4⋊Dic73—41no—
89#11924007—44no—
90C22⋊Dic73—44no—
91#11924003—22no—
92#11924003—41no—
93C24×C73—41no—
94C24×C73D811yescharacteristic
95C2×D4×C73—22no—
96C2×D4×C73—44no—
97C2×D4×C73—41no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—82no—
103C2×D4×C73—82no—
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73—21no—
108C22×C4×C73—21no—
109C22×C4×C73—21no—
110C22×C4×C73—41no—
111C22×C4×C73D811yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113D8×C73—22no—
114D8×C73—22no—
115D8×C73—41no—
116D8×C73—41no—
117C2×C8×C73—21no—
118C2×C8×C73—21no—
119C2×C8×C73—21no—
120C2×C8×C73—41no—
121C2×C8×C73—41no—
122C2×C8×C73—41no—
123C2×C8×C73—41no—
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73C2×D411yescharacteristic
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2×D411yescharacteristic
128C2×C8×C73C2×D411yescharacteristic
129C2×C8×C73#4111yescharacteristic
130C4⋊C4×C73—22no—
131C4⋊C4×C73—41no—
132C22⋊C4×C73—44no—
133C23×C73—21no—
134C23×C73—21no—
135C23×C73—21no—
136C23×C73—21no—
137C23×C73—21no—
138C23×C73—21no—
139C23×C73—42no—
140C23×C73—81no—
141C23×C73—81no—
142C23×C73—81no—
143C23×C73—81no—
144C23×C73—81no—
145C23×C73—81no—
146C23×C73SD3212yescomplemented
147C23×C73C2×D811yescharacteristic
148D4×C73—42no—
149D4×C73—42no—
150D4×C73—88no—
151D4×C73—81no—
152D4×C73—81no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—42no—
161C2×C4×C73—81no—
162C2×C4×C73—81no—
163C2×C4×C73#12011yescharacteristic
164C2×C4×C73#12011yescharacteristic
165C2×C4×C73#12111yescharacteristic
166C2×C4×C73#13511yescharacteristic
167C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
169C2×Dic73—42no—
170C2×Dic73—42no—
171C2×Dic73—81no—
172C2×Dic73—81no—
173C2×Dic73—81no—
174C2×Dic73—81no—
175Dic146—42no—
176Dic146—42no—
177Dic146—81no—
178Dic146—81no—
179C8×C73—41no—
180C8×C73#12011yescharacteristic
181C8×C73#12111yescharacteristic
182C8×C73#12111yescharacteristic
183C8×C73#12711yescharacteristic
184C22×C73—22no—
185C22×C73—22no—
186C22×C73—22no—
187C22×C73—41no—
188C22×C73—41no—
189C22×C73—41no—
190C22×C73—41no—
191C22×C73—82no—
192C22×C73—82no—
193C22×C73—82no—
194C22×C73—161no—
195C22×C73—161no—
196C22×C73—161no—
197C22×C73—161no—
198C22×C73#44711yescharacteristic
199C22×C73#44711yescharacteristic
200C22×C73#46611yescharacteristic
201C22×C73C2×SD3211yescharacteristic
202C22×C73#50811yescharacteristic
203C22×C73#50911yescharacteristic
204C22×C73#51011yescharacteristic
205C4×C73—41no—
206C4×C73—82no—
207C4×C73#39211yescharacteristic
208C4×C73#44911yescharacteristic
209C4×C73#44911yescharacteristic
210C4×C73#46911yescharacteristic
211Dic73—82no—
212Dic73—82no—
213#13305—731nomaximal, Sylow
214C2×C73—42no—
215C2×C73—162no—
216C2×C73#199911yescharacteristic
217C2×C73#218511yescharacteristic
218C2×C73#218611yescharacteristic
219C2×C73#218711yescharacteristic
220C2×C73#218911yescharacteristic
221C2×C73#221711yescharacteristic
222C2×C73#222011yescharacteristic
223C22×SD32—731no—
224C2×#467—731no—
225C2×#466D7311yescharacteristic
226C2×#358—731no—
227C2×#357—731no—
228C2×C22⋊C16—731no—
229#1380—731no—
230C73#1330511yescharacteristic, complemented, Sylow
231C22×Q16—731no—
232C22×D8C2×D7311yescharacteristic
233C23×C8C2×D7311yescharacteristic
234C2×C22⋊Q8—1461no—
235C2×#121—21no—
236C2×SD32—1462no—
237C2×SD32—1462no—
238C2×SD32—1462no—
239C22×C16—731no—
240C22×C16—731no—
241#467—1462no—
242#467—1462no—
243#466—22no—
244#466—22no—
245C2×#107—731no—
246C2×#107C2×D7311yescharacteristic
247C2×#103—1461no—
248C2×#102—21no—
249#358—1462no—
250#357—1462no—
251C22⋊C16—1462no—
252C22×Q8—1461no—
253C22×D4—21no—
254C22×D4—41no—
255C23×C4C22⋊D7311yescharacteristic
256C2×Q16—732no—
257C2×Q16—1461no—
258C2×Q16—1461no—
259C2×Q16—1461no—
260C2×Q16—1461no—
261C2×Q16—1461no—
262C2×D8—21no—
263C2×D8—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8—21no—
267C2×D8C22⋊D7312yes—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8C22⋊D7311yescharacteristic
275C22×C8C22×D7311yescharacteristic
276C22⋊Q8—2922no—
277C22⋊Q8—1464no—
278#121—24no—
279#121—42no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—2921no—
282C2×C4⋊C4—1461no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—2921no—
285SD32—1468no—
286C2×C16—732no—
287C2×C16—1461no—
288C2×C16—1461no—
289C2×C16—1461no—
290C2×C16—1461no—
291C2×C16—1461no—
292#107—21no—
293#107—732no—
294#107—1461no—
295#107C22⋊D7312yes—
296#103—1464no—
297#102—24no—
298C24—41no—
299C24#1192401311yescharacteristic
300C2×Q8—2921no—
301C2×Q8—2921no—
302C2×Q8—2921no—
303C2×Q8—2921no—
304C2×Q8—2921no—
305C2×Q8—1462no—
306C2×D4—22no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C2×D4—82no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—2921no—
323C22×C4—2921no—
324C22×C4#1192401311yescharacteristic
325C22×C4C2×C22⋊D7311yescharacteristic
326Q16—2921no—
327Q16—2921no—
328Q16—1462no—
329Q16—1462no—
330D8—22no—
331D8—22no—
332D8—41no—
333D8—41no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8D4×D7311yescharacteristic
342C2×C8D4×D7311yescharacteristic
343C2×C8D4×D7311yescharacteristic
344C2×C8C2×C22⋊D7311yescharacteristic
345C2×C8C2×C22⋊D7311yescharacteristic
346C2×C8#1192402711yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—2924no—
350C4⋊C4—2921no—
351C4⋊C4—1462no—
352C22⋊C4—44no—
353C22⋊C4—2924no—
354C16—2921no—
355C16—1462no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#1455653512yescomplemented
370C23C2×#1192401311yescharacteristic
371Q8—2922no—
372Q8—2922no—
373Q8—5841no—
374Q8—5841no—
375D4—42no—
376D4—42no—
377D4—88no—
378D4—81no—
379D4—81no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—2922no—
391C2×C4—2922no—
392C2×C4—5841no—
393C2×C4—5841no—
394C2×C4—5841no—
395C2×C4—5841no—
396C2×C4#1455655911yescharacteristic
397C2×C4#1455656011yescharacteristic
398C2×C4#1455656311yescharacteristic
399C2×C4#1455661111yescharacteristic
400C2×C4#1455661611yescharacteristic
401C2×C4#1455661711yescharacteristic
402C8—41no—
403C8#1455655911yescharacteristic
404C8#1455656211yescharacteristic
405C8#1455656311yescharacteristic
406C8#1455658711yescharacteristic
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22#1798872011yescharacteristic
422C22#1798872111yescharacteristic
423C22#1798882111yescharacteristic
424C22C2×#1455653511yescharacteristic
425C22#1798901711yescharacteristic
426C22#1798902311yescharacteristic
427C22#1798902711yescharacteristic
428C4—41no—
429C4—82no—
430C4—5842no—
431C4—5842no—
432C4#1798849511yescharacteristic
433C4#1798873011yescharacteristic
434C4#1798873211yescharacteristic
435C4#1798883811yescharacteristic
436C2—42no—
437C2—162no—
438C2#2261092711yescharacteristic
439C2#2261184711yescharacteristic
440C2#2261185211yescharacteristic
441C2#2261186011yescharacteristic
442C2#2261187111yescharacteristic
443C2#2261202511yescharacteristic
444C2#2261204211yescharacteristic
subgroupG/H
centerC23C2×#11924013
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
Fitting, prime radicalC2×#466×C73C2
socleC23×C73C2×D8
hypercenterC2×#466D73
hypocenterC73#13305
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#14556535C2semidirect
2. C2×#17988828C2semidirect
3. C2×#17988390C2semidirect
4. C2×#17988388C2semidirect
5. C2×C22⋊#11924003C2semidirect
6. #22608022C2semidirect
35. C2×Dic292D4semidirect
47. C2×#11924003D4semidirect
49. #14556525D4semidirect
146. C23×C73SD32semidirect
230. C73#13305semidirect
369. C23#14556535semidirect

Maximal subgroups

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

subgroupindex
C22×#145565352
C2×#179888282
C2×#179883902
C2×#179883882
C2×C22⋊#119240032
#226080222
C2×#466×C732
#1330573