Finite Group Databasebeta

← #31916154

Subgroups — #31916154

Order 18176.

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×#14556072C211yesmaximal, characteristic, complemented
2C2×#17986240C211yesmaximal, characteristic, complemented
3C2×#17985802C211yesmaximal, characteristic, complemented
4C2×#17985800C211yesmaximal, characteristic, complemented
5C2×C22⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
6#22586431C211yesmaximal, characteristic, complemented
7C2×#466×C71C211yesmaximal, characteristic
8C22×Dic284C2211yescharacteristic
9C2×#14556107—21no—
10C2×#14556072—22no—
11C2×#14556072—22no—
12C2×#14556072—22no—
13C22×C71⋊C16C2211yescharacteristic
14C22×C71⋊C16C2211yescharacteristic
15#17986240—22no—
16#17986240—22no—
17C2×#14556062C2211yescharacteristic
18C2×#14556053—21no—
19#17985802—22no—
20#17985800—22no—
21C22⋊(C71⋊C16)—22no—
22C22×D8×C71C2211yescharacteristic
23C23×C8×C71C2211yescharacteristic
24C2×#121×C71—21no—
25#466×C71—22no—
26#466×C71—22no—
27C2×#107×C71C2211yescharacteristic
28C2×#102×C71—21no—
29C22×Dic142—21no—
30C2×Dic284—21no—
31C2×Dic284—21no—
32C2×Dic284—21no—
33C2×Dic284—21no—
34C2×Dic284—21no—
35C2×Dic284D412yescomplemented
36#14556107—24no—
37#14556107—42no—
38C2×C4⋊Dic71—21no—
39C2×Dic71⋊C4—41no—
40C2×C22⋊Dic71—41no—
41#14556072—28no—
42C2×C71⋊C16—21no—
43C2×C71⋊C16—21no—
44C2×C71⋊C16—21no—
45C2×C71⋊C16—21no—
46C2×C71⋊C16—21no—
47C2×C71⋊C16D412yescomplemented
48#14556062—21no—
49#14556062D412yescomplemented
50#14556053—24no—
51C22×D4×C71—21no—
52C22×D4×C71—41no—
53C23×C4×C71D411yescharacteristic
54C2×D8×C71—21no—
55C2×D8×C71—21no—
56C2×D8×C71—21no—
57C2×D8×C71—21no—
58C2×D8×C71—21no—
59C2×D8×C71D412yes—
60C22×C8×C71—21no—
61C22×C8×C71—21no—
62C22×C8×C71—21no—
63C22×C8×C71—21no—
64C22×C8×C71—21no—
65C22×C8×C71—21no—
66C22×C8×C71D411yescharacteristic
67C22×C8×C71C2311yescharacteristic
68#121×C71—24no—
69#121×C71—42no—
70C2×C4⋊C4×C71—21no—
71C2×C22⋊C4×C71—41no—
72#107×C71—21no—
73#107×C71D412yes—
74#102×C71—24no—
75C2×Dic142—22no—
76C2×Dic142—41no—
77C2×Dic142—41no—
78C2×Dic142—41no—
79C2×Dic142—41no—
80C2×Dic142—41no—
81C22×Dic71—41no—
82C22×Dic71—41no—
83Dic284—22no—
84Dic284—22no—
85Dic284—41no—
86Dic284—41no—
87C4⋊Dic71—22no—
88C4⋊Dic71—41no—
89Dic71⋊C4—44no—
90C22⋊Dic71—44no—
91C71⋊C16—22no—
92C71⋊C16—41no—
93C24×C71—41no—
94C24×C71D811yescharacteristic
95C2×D4×C71—22no—
96C2×D4×C71—44no—
97C2×D4×C71—41no—
98C2×D4×C71—41no—
99C2×D4×C71—41no—
100C2×D4×C71—41no—
101C2×D4×C71—41no—
102C2×D4×C71—82no—
103C2×D4×C71—82no—
104C22×C4×C71—21no—
105C22×C4×C71—21no—
106C22×C4×C71—21no—
107C22×C4×C71—21no—
108C22×C4×C71—21no—
109C22×C4×C71—21no—
110C22×C4×C71—41no—
111C22×C4×C71D811yescharacteristic
112C22×C4×C71C2×D411yescharacteristic
113D8×C71—22no—
114D8×C71—22no—
115D8×C71—41no—
116D8×C71—41no—
117C2×C8×C71—21no—
118C2×C8×C71—21no—
119C2×C8×C71—21no—
120C2×C8×C71—41no—
121C2×C8×C71—41no—
122C2×C8×C71—41no—
123C2×C8×C71—41no—
124C2×C8×C71C2×D411yescharacteristic
125C2×C8×C71C2×D411yescharacteristic
126C2×C8×C71C2×D411yescharacteristic
127C2×C8×C71C2×D411yescharacteristic
128C2×C8×C71C2×D411yescharacteristic
129C2×C8×C71#4111yescharacteristic
130C4⋊C4×C71—22no—
131C4⋊C4×C71—41no—
132C22⋊C4×C71—44no—
133C23×C71—21no—
134C23×C71—21no—
135C23×C71—21no—
136C23×C71—21no—
137C23×C71—21no—
138C23×C71—21no—
139C23×C71—42no—
140C23×C71—81no—
141C23×C71—81no—
142C23×C71—81no—
143C23×C71—81no—
144C23×C71—81no—
145C23×C71—81no—
146C23×C71SD3212yescomplemented
147C23×C71C2×D811yescharacteristic
148D4×C71—42no—
149D4×C71—42no—
150D4×C71—88no—
151D4×C71—81no—
152D4×C71—81no—
153C2×C4×C71—21no—
154C2×C4×C71—21no—
155C2×C4×C71—21no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—41no—
160C2×C4×C71—42no—
161C2×C4×C71—81no—
162C2×C4×C71—81no—
163C2×C4×C71#12011yescharacteristic
164C2×C4×C71#12011yescharacteristic
165C2×C4×C71#12111yescharacteristic
166C2×C4×C71#13511yescharacteristic
167C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
169C2×Dic71—42no—
170C2×Dic71—42no—
171C2×Dic71—81no—
172C2×Dic71—81no—
173C2×Dic71—81no—
174C2×Dic71—81no—
175Dic142—42no—
176Dic142—42no—
177Dic142—81no—
178Dic142—81no—
179C8×C71—41no—
180C8×C71#12011yescharacteristic
181C8×C71#12111yescharacteristic
182C8×C71#12111yescharacteristic
183C8×C71#12711yescharacteristic
184C22×C71—22no—
185C22×C71—22no—
186C22×C71—22no—
187C22×C71—41no—
188C22×C71—41no—
189C22×C71—41no—
190C22×C71—41no—
191C22×C71—82no—
192C22×C71—82no—
193C22×C71—82no—
194C22×C71—161no—
195C22×C71—161no—
196C22×C71—161no—
197C22×C71—161no—
198C22×C71#44711yescharacteristic
199C22×C71#44711yescharacteristic
200C22×C71#46611yescharacteristic
201C22×C71C2×SD3211yescharacteristic
202C22×C71#50811yescharacteristic
203C22×C71#50911yescharacteristic
204C22×C71#51011yescharacteristic
205C4×C71—41no—
206C4×C71—82no—
207C4×C71#39211yescharacteristic
208C4×C71#44911yescharacteristic
209C4×C71#44911yescharacteristic
210C4×C71#46911yescharacteristic
211Dic71—82no—
212Dic71—82no—
213#13305—711nomaximal, Sylow
214C2×C71—42no—
215C2×C71—162no—
216C2×C71#199911yescharacteristic
217C2×C71#218511yescharacteristic
218C2×C71#218611yescharacteristic
219C2×C71#218711yescharacteristic
220C2×C71#218911yescharacteristic
221C2×C71#221711yescharacteristic
222C2×C71#222011yescharacteristic
223C22×SD32—711no—
224C2×#467—711no—
225C2×#466D7111yescharacteristic
226C2×#358—711no—
227C2×#357—711no—
228C2×C22⋊C16—711no—
229#1380—711no—
230C71#1330511yescharacteristic, complemented, Sylow
231C22×Q16—711no—
232C22×D8C2×D7111yescharacteristic
233C23×C8C2×D7111yescharacteristic
234C2×C22⋊Q8—1421no—
235C2×#121—21no—
236C2×SD32—1422no—
237C2×SD32—1422no—
238C2×SD32—1422no—
239C22×C16—711no—
240C22×C16—711no—
241#467—1422no—
242#467—1422no—
243#466—22no—
244#466—22no—
245C2×#107—711no—
246C2×#107C2×D7111yescharacteristic
247C2×#103—1421no—
248C2×#102—21no—
249#358—1422no—
250#357—1422no—
251C22⋊C16—1422no—
252C22×Q8—1421no—
253C22×D4—21no—
254C22×D4—41no—
255C23×C4C22⋊D7111yescharacteristic
256C2×Q16—712no—
257C2×Q16—1421no—
258C2×Q16—1421no—
259C2×Q16—1421no—
260C2×Q16—1421no—
261C2×Q16—1421no—
262C2×D8—21no—
263C2×D8—21no—
264C2×D8—21no—
265C2×D8—21no—
266C2×D8—21no—
267C2×D8C22⋊D7112yes—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—21no—
274C22×C8C22⋊D7111yescharacteristic
275C22×C8C22×D7111yescharacteristic
276C22⋊Q8—2842no—
277C22⋊Q8—1424no—
278#121—24no—
279#121—42no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—2841no—
282C2×C4⋊C4—1421no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—2841no—
285SD32—1428no—
286C2×C16—712no—
287C2×C16—1421no—
288C2×C16—1421no—
289C2×C16—1421no—
290C2×C16—1421no—
291C2×C16—1421no—
292#107—21no—
293#107—712no—
294#107—1421no—
295#107C22⋊D7112yes—
296#103—1424no—
297#102—24no—
298C24—41no—
299C24#1176585011yescharacteristic
300C2×Q8—2841no—
301C2×Q8—2841no—
302C2×Q8—2841no—
303C2×Q8—2841no—
304C2×Q8—2841no—
305C2×Q8—1422no—
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—2841no—
323C22×C4—2841no—
324C22×C4#1176585011yescharacteristic
325C22×C4C2×C22⋊D7111yescharacteristic
326Q16—2841no—
327Q16—2841no—
328Q16—1422no—
329Q16—1422no—
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×D7111yescharacteristic
342C2×C8D4×D7111yescharacteristic
343C2×C8D4×D7111yescharacteristic
344C2×C8C2×C22⋊D7111yescharacteristic
345C2×C8C2×C22⋊D7111yescharacteristic
346C2×C8#1176586411yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—2844no—
350C4⋊C4—2841no—
351C4⋊C4—1422no—
352C22⋊C4—44no—
353C22⋊C4—2844no—
354C16—2841no—
355C16—1422no—
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#1455607212yescomplemented
370C23C2×#1176585011yescharacteristic
371Q8—2842no—
372Q8—2842no—
373Q8—5681no—
374Q8—5681no—
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—2842no—
391C2×C4—2842no—
392C2×C4—5681no—
393C2×C4—5681no—
394C2×C4—5681no—
395C2×C4—5681no—
396C2×C4#1455609611yescharacteristic
397C2×C4#1455609711yescharacteristic
398C2×C4#1455610011yescharacteristic
399C2×C4#1455614811yescharacteristic
400C2×C4#1455615311yescharacteristic
401C2×C4#1455615411yescharacteristic
402C8—41no—
403C8#1455609611yescharacteristic
404C8#1455609911yescharacteristic
405C8#1455610011yescharacteristic
406C8#1455612411yescharacteristic
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#1798613211yescharacteristic
422C22#1798613311yescharacteristic
423C22#1798623311yescharacteristic
424C22C2×#1455607211yescharacteristic
425C22#1798642911yescharacteristic
426C22#1798643511yescharacteristic
427C22#1798643911yescharacteristic
428C4—41no—
429C4—82no—
430C4—5682no—
431C4—5682no—
432C4#1798590711yescharacteristic
433C4#1798614211yescharacteristic
434C4#1798614411yescharacteristic
435C4#1798625011yescharacteristic
436C2—42no—
437C2—162no—
438C2#2258933611yescharacteristic
439C2#2259025611yescharacteristic
440C2#2259026111yescharacteristic
441C2#2259026911yescharacteristic
442C2#2259028011yescharacteristic
443C2#2259043411yescharacteristic
444C2#2259045111yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
Fitting, prime radicalC2×#466×C71C2
socleC23×C71C2×D8
hypercenterC2×#466D71
hypocenterC71#13305
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#14556072C2semidirect
2. C2×#17986240C2semidirect
3. C2×#17985802C2semidirect
4. C2×#17985800C2semidirect
5. C2×C22⋊(C71⋊C16)C2semidirect
6. #22586431C2semidirect
35. C2×Dic284D4semidirect
47. C2×C71⋊C16D4semidirect
49. #14556062D4semidirect
146. C23×C71SD32semidirect
230. C71#13305semidirect
369. C23#14556072semidirect

Maximal subgroups

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

subgroupindex
C22×#145560722
C2×#179862402
C2×#179858022
C2×#179858002
C2×C22⋊(C71⋊C16)2
#225864312
C2×#466×C712
#1330571