Finite Group Databasebeta

← #36361860

Subgroups — #36361860

Order 22144.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149665C211yesmaximal, characteristic, complemented
2C2×#19149656C211yesmaximal, characteristic, complemented
3C2×#19149651C211yesmaximal, characteristic
4C22×C4×Dic173C211yesmaximal, characteristic, complemented
5#26011395C211yesmaximal, characteristic, complemented
6#26011391C211yesmaximal, characteristic, complemented
7#26011384C211yesmaximal, characteristic, complemented
8#26011383C211yesmaximal, characteristic, complemented
9#26011380C211yesmaximal, characteristic, complemented
10#26011379C211yesmaximal, characteristic, complemented
11#26011371C211yesmaximal, characteristic, complemented
12#26011370C211yesmaximal, characteristic, complemented
13#26011368C211yesmaximal, characteristic, complemented
14#26011355C211yesmaximal, characteristic, complemented
15#386×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C22×C22⋊D173C2211yescharacteristic
19C23×Dic173C2211yescharacteristic
20C23×Dic173C2211yescharacteristic
21#19149665—24no—
22#19149656—24no—
23#19149651—24no—
24C2×C4⋊Dic173C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C22⋊Dic173C2211yescharacteristic
30C2×C22⋊Dic173C2211yescharacteristic
31C2×C22⋊Dic173C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×C4×Dic173—24no—
38C2×C4×Dic173C2211yescharacteristic
39C2×C4×Dic173C2211yescharacteristic
40C2×C4×Dic173C2211yescharacteristic
41C2×C4×Dic173C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47#19149586C2211yescharacteristic
48C22×D4×C173C2211yescharacteristic
49C23×C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52C2×C22⋊C4×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54#95×C173C2211yescharacteristic
55C23×D173C2311yescharacteristic
56C2×Dic346—22no—
57C2×Dic346—22no—
58C2×Dic346—22no—
59C2×C22⋊D173—22no—
60C2×C22⋊D173—22no—
61C2×C22⋊D173—22no—
62C2×C22⋊D173—22no—
63C2×C22⋊D173—22no—
64C2×C22⋊D173—22no—
65C22×Dic173—22no—
66C22×Dic173—22no—
67C22×Dic173—22no—
68C22×Dic173—22no—
69C22×Dic173—22no—
70C22×Dic173D414yes—
71C22×Dic173C2311yescharacteristic
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C22×Dic173C2311yescharacteristic
77C22×Dic173C2311yescharacteristic
78C4⋊Dic173—22no—
79#14583805—22no—
80#14583805—22no—
81#14583805—22no—
82C22⋊Dic173—22no—
83C22⋊Dic173—22no—
84C22⋊Dic173—22no—
85C22⋊Dic173—22no—
86#14583803—22no—
87#14583803—22no—
88#14583803—22no—
89#14583803—22no—
90#14583803—22no—
91C4×Dic173—22no—
92C4×Dic173—22no—
93C4×Dic173—22no—
94C4×Dic173D414yescomplemented
95C24×C173C2311yescharacteristic
96C24×C173C2311yescharacteristic
97C2×D4×C173—22no—
98C2×D4×C173—22no—
99C2×D4×C173—22no—
100C22×C4×C173—22no—
101C22×C4×C173—22no—
102C22×C4×C173—22no—
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108C22⋊C4×C173—22no—
109C22⋊C4×C173—22no—
110C22⋊C4×C173—22no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—21no—
118Dic346—44no—
119C22⋊D173—44no—
120C22⋊D173—44no—
121C2×Dic173—28no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—28no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173C2×D412yes—
139C2×Dic173C2×D412yes—
140C2×Dic173C2×D412yes—
141C2×Dic173C2×D412yes—
142C2×Dic173#4112yes—
143C2×Dic173#4112yes—
144C23×C173—21no—
145C23×C173—21no—
146C23×C173—21no—
147C23×C173—21no—
148C23×C173—21no—
149C23×C173—21no—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173C2×D412yes—
156C23×C173#4112yes—
157C23×C173#4112yes—
158C23×C173C2411yescharacteristic
159D4×C173—44no—
160C2×C4×C173—28no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173C2×D412yes—
175C2×C4×C173#4112yes—
176C22×C173—22no—
177C22×C173—22no—
178C22×C173—22no—
179C22×C173—22no—
180C22×C173—22no—
181C22×C173—22no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173#12414yescomplemented
190C22×C173C22×D411yescharacteristic
191C22×C173C22×D411yescharacteristic
192C22×C173C2×#4111yescharacteristic
193C22×C173C2×#4111yescharacteristic
194C22×C173C2×#4111yescharacteristic
195C22×C173C2×#4111yescharacteristic
196C22×C173C2×#4111yescharacteristic
197C2×D173—41no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173—41no—
203C2×D173—41no—
204C4×C173—22no—
205C4×C173—22no—
206C4×C173—41no—
207C4×C173—41no—
208C4×C173—41no—
209Dic173—22no—
210Dic173—22no—
211Dic173—22no—
212Dic173—41no—
213Dic173—41no—
214Dic173—41no—
215Dic173#12114yes—
216C2×C173—24no—
217C2×C173—81no—
218C2×C173C2×#12111yescharacteristic
219C2×C173#52511yescharacteristic
220C2×C173#52511yescharacteristic
221C2×C173C2×#12411yescharacteristic
222C2×C173#52911yescharacteristic
223C2×C173#52911yescharacteristic
224C2×C173#53211yescharacteristic
225D173—81no—
226C173#285011yescharacteristic, complemented, Sylow
227#2850—1731nomaximal, Sylow
228C2×#124—1731no—
229C2×C22⋊Q8—1731no—
230C2×#121—1731no—
231C22×C42—1731no—
232#391—1731no—
233#390—1731no—
234#388—1731no—
235#388—1731no—
236#388—1731no—
237#388—1731no—
238#386—1731no—
239#386—1731no—
240#386—1731no—
241#386D17311yescharacteristic, complemented
242#382—1731no—
243C22×Q8—1731no—
244C22×D4—1731no—
245C22×D4—1731no—
246C22×D4C2×D17311yescharacteristic
247C23×C4—1731no—
248C23×C4—1731no—
249C23×C4C2×D17311yescharacteristic
250#124—3464no—
251C22⋊Q8—3464no—
252#121—3464no—
253C2×C4⋊C4—1731no—
254C2×C4⋊C4—1731no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4—1731no—
257C2×C22⋊C4—1731no—
258C2×C22⋊C4—1731no—
259C2×C22⋊C4—1731no—
260C2×C22⋊C4—1731no—
261C2×C22⋊C4—1731no—
262C2×C22⋊C4—1731no—
263C2×C22⋊C4—1731no—
264C2×C22⋊C4—1731no—
265C2×C22⋊C4—1731no—
266C2×C22⋊C4C2×D17311yescharacteristic
267C2×C22⋊C4C2×D17311yescharacteristic
268C2×C22⋊C4C2×D17311yescharacteristic
269C2×C42—3464no—
270C2×C42—1731no—
271C2×C42—1731no—
272C2×C42—1731no—
273C2×C42—1731no—
274#95—1731no—
275#95—1731no—
276#95—1731no—
277#95—1731no—
278#95—1731no—
279#95—1731no—
280#95C2×D17311yescharacteristic
281#95C2×D17311yescharacteristic
282C24—1731no—
283C24C22×D17311yescharacteristic
284C24C22×D17311yescharacteristic
285C2×Q8—3462no—
286C2×Q8—3462no—
287C2×Q8—3462no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—3462no—
292C2×D4—3462no—
293C2×D4—3462no—
294C2×D4—3462no—
295C2×D4—3462no—
296C2×D4—3462no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—3462no—
301C22×C4—3462no—
302C22×C4—3462no—
303C22×C4—3462no—
304C22×C4—3462no—
305C22×C4—1731no—
306C22×C4—1731no—
307C22×C4—1731no—
308C22×C4—1734no—
309C22×C4—1731no—
310C22×C4—1731no—
311C22×C4—1731no—
312C22×C4—1731no—
313C22×C4C22×D17311yescharacteristic
314C22×C4C22×D17311yescharacteristic
315C22×C4C22×D17311yescharacteristic
316C22×C4C22×D17311yescharacteristic
317C22×C4C22×D17311yescharacteristic
318C4⋊C4—3462no—
319C4⋊C4—3462no—
320C4⋊C4—3462no—
321C4⋊C4—3462no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—3462no—
326C22⋊C4—3462no—
327C22⋊C4—3462no—
328C22⋊C4—3462no—
329C22⋊C4—3462no—
330C22⋊C4—3462no—
331C22⋊C4—3462no—
332C22⋊C4—3462no—
333C22⋊C4—3462no—
334C42—3462no—
335C42—3462no—
336C42—3462no—
337C42—1734no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—3461no—
350C23—3461no—
351C23—3461no—
352C23—3461no—
353C23—3461no—
354C23—3461no—
355C23—3461no—
356C23D4×D17312yes—
357C23#1458382512yes—
358C23#1458382612yes—
359C23C23×D17311yescharacteristic
360Q8—6924no—
361D4—44no—
362D4—6924no—
363D4—6924no—
364C2×C4—28no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—3468no—
379C2×C4—3461no—
380C2×C4—3461no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—3468no—
386C2×C4—3461no—
387C2×C4—3461no—
388C2×C4—3461no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—1732no—
396C2×C4—1732no—
397C2×C4—1732no—
398C2×C4—1732no—
399C2×C4—1732no—
400C2×C4—1732no—
401C2×C4D4×D17312yes—
402C2×C4#1458382512yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—6921no—
417C22—6921no—
418C22—6921no—
419C22—6921no—
420C22—6921no—
421C22—6921no—
422C22—6921no—
423C22#1914966514yescomplemented
424C22C2×D4×D17311yescharacteristic
425C22C2×D4×D17311yescharacteristic
426C22#41×D17311yescharacteristic
427C22#41×D17311yescharacteristic
428C22C2×#1458382511yescharacteristic
429C22C2×#1458382511yescharacteristic
430C22C2×#1458382611yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—3462no—
437C4—3462no—
438C4—3462no—
439C4—1734no—
440C4—6921no—
441C4—6921no—
442C4—6921no—
443C2—24no—
444C2—81no—
445C2—13841no—
446C2#121×D17311yescharacteristic
447C2#2601199711yescharacteristic
448C2#2601200011yescharacteristic
449C2C2×#1914966511yescharacteristic
450C2#2601201811yescharacteristic
451C2#2601202111yescharacteristic
452C2#2601203611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#386×C173C2
prime radicalC2×#19149651C2
hypercenter#386D173
hypocenterC173#2850
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#19149665C2semidirect
2. C2×#19149656C2semidirect
4. C22×C4×Dic173C2semidirect
5. #26011395C2semidirect
6. #26011391C2semidirect
7. #26011384C2semidirect
8. #26011383C2semidirect
9. #26011380C2semidirect
10. #26011379C2semidirect
11. #26011371C2semidirect
12. #26011370C2semidirect
13. #26011368C2semidirect
14. #26011355C2semidirect
15. #386×C173C2semidirect
94. C4×Dic173D4semidirect
189. C22×C173#124semidirect
226. C173#2850semidirect
241. #386D173semidirect
423. C22#19149665semidirect

Maximal subgroups

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

subgroupindex
C2×#191496652
C2×#191496562
C2×#191496512
C22×C4×Dic1732
#260113952
#260113912
#260113842
#260113832
#260113802
#260113792
#260113712
#260113702
#260113682
#260113552
#386×C1732
#2850173