Finite Group Databasebeta

← #36356389

Subgroups — #36356389

Order 22144.

928 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14583825C211yesmaximal, characteristic
2C2×C4×C22⋊D173C211yesmaximal, characteristic, complemented
3C2×#19149639C211yesmaximal, characteristic, complemented
4#26011395C211yesmaximal, characteristic, complemented
5#26011395C211yesmaximal, characteristic, complemented
6#26011371C211yesmaximal, characteristic, complemented
7#26011367C211yesmaximal, characteristic, complemented
8#26011367C211yesmaximal, characteristic, complemented
9#26011353C211yesmaximal, characteristic, complemented
10#26011352C211yesmaximal, characteristic, complemented
11#26011349C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
13C2×#19149586C211yesmaximal, characteristic, complemented
14#95×D173C211yesmaximal, characteristic, complemented
15#386×C173C211yesmaximal, characteristic, complemented
16C2×#14583825—24no—
17C2×#14583825—24no—
18C2×#14583825C418yescomplemented
19C22×Dic346C2211yescharacteristic
20C22×C22⋊D173C2211yescharacteristic
21C22×C22⋊D173C2211yescharacteristic
22C23×Dic173C2211yescharacteristic
23C23×Dic173C2211yescharacteristic
24C23×Dic173C2211yescharacteristic
25C22×C4×D173C2211yescharacteristic
26C22×C4×D173C2211yescharacteristic
27C22×C4×D173C2211yescharacteristic
28C4×C22⋊D173—24no—
29#19149639—24no—
30C2×#14583805C2211yescharacteristic
31C2×#14583805C2211yescharacteristic
32C2×C22⋊Dic173C2211yescharacteristic
33C2×C22⋊Dic173C2211yescharacteristic
34C2×C22⋊Dic173C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×C4×Dic173C2211yescharacteristic
38C2×C4×Dic173C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47#19149586C2211yescharacteristic
48#19149586C2211yescharacteristic
49C22×D4×C173C2211yescharacteristic
50C23×C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52C2×C22⋊C4×C173C2211yescharacteristic
53C2×C22⋊C4×C173C2211yescharacteristic
54#95×C173C2211yescharacteristic
55#95×C173C2211yescharacteristic
56C23×D173C2311yescharacteristic
57#14583825—232no—
58C2×Dic346—22no—
59C2×Dic346—22no—
60C2×Dic346C2×C414yescomplemented
61C2×C22⋊D173—22no—
62C2×C22⋊D173—22no—
63C2×C22⋊D173C2×C414yes—
64C2×C22⋊D173C2×C414yescomplemented
65C2×C22⋊D173D414yes—
66C2×C22⋊D173D414yes—
67C22×Dic173—22no—
68C22×Dic173—22no—
69C22×Dic173—22no—
70C22×Dic173—22no—
71C22×Dic173—22no—
72C22×Dic173—22no—
73C22×Dic173—41no—
74C22×Dic173C2×C414yescomplemented
75C22×Dic173C2×C414yescomplemented
76C22×Dic173C2311yescharacteristic
77C22×Dic173C2311yescharacteristic
78C22×Dic173C2311yescharacteristic
79C22×Dic173C2311yescharacteristic
80C22×Dic173C2311yescharacteristic
81C22×Dic173C2311yescharacteristic
82C22×Dic173C2311yescharacteristic
83C2×C4×D173—22no—
84C2×C4×D173—22no—
85C2×C4×D173—22no—
86C2×C4×D173—22no—
87C2×C4×D173—22no—
88C2×C4×D173—22no—
89C2×C4×D173—41no—
90C2×C4×D173—41no—
91C2×C4×D173C2×C414yescomplemented
92#14583805—22no—
93#14583805—22no—
94C22⋊Dic173—22no—
95C22⋊Dic173—22no—
96C22⋊Dic173—41no—
97#14583803—22no—
98#14583803—22no—
99C4×Dic173—22no—
100C4×Dic173—22no—
101C24×C173C2311yescharacteristic
102C24×C173C2311yescharacteristic
103C2×D4×C173—22no—
104C2×D4×C173—22no—
105C2×D4×C173C2×C414yescomplemented
106C22×C4×C173—22no—
107C22×C4×C173—22no—
108C22×C4×C173—41no—
109C22×C4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113C22×C4×C173C2311yescharacteristic
114C22⋊C4×C173—22no—
115C22⋊C4×C173—22no—
116C22⋊C4×C173—41no—
117C22×D173—21no—
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173C22×C412yes—
122C22×D173C2×D412yes—
123C22×D173C2×D412yes—
124Dic346—28no—
125C22⋊D173—28no—
126C22⋊D173C22⋊C4116yescomplemented
127C2×Dic173—28no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—28no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—44no—
145C2×Dic173C22×C412yes—
146C2×Dic173C22×C412yes—
147C2×Dic173C22×C412yes—
148C2×Dic173C2×D412yes—
149C2×Dic173C2×D412yes—
150C2×Dic173#4112yescomplemented
151C4×D173—28no—
152C4×D173—44no—
153C4×D173—44no—
154C23×C173—21no—
155C23×C173—21no—
156C23×C173—21no—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—21no—
160C23×C173—21no—
161C23×C173—21no—
162C23×C173—21no—
163C23×C173—21no—
164C23×C173C22×C412yes—
165C23×C173C22×C412yes—
166C23×C173C2×D412yes—
167C23×C173C2×D412yes—
168C23×C173C2411yescharacteristic
169D4×C173—28no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—21no—
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173—21no—
177C2×C4×C173—21no—
178C2×C4×C173—21no—
179C2×C4×C173—21no—
180C2×C4×C173—21no—
181C2×C4×C173—21no—
182C2×C4×C173—21no—
183C2×C4×C173—44no—
184C2×C4×C173C22×C412yes—
185C2×C4×C173#4112yescomplemented
186C22×C173—22no—
187C22×C173—22no—
188C22×C173—22no—
189C22×C173—22no—
190C22×C173—22no—
191C22×C173—22no—
192C22×C173—22no—
193C22×C173—22no—
194C22×C173—22no—
195C22×C173—41no—
196C22×C173—41no—
197C22×C173—41no—
198C22×C173—41no—
199C22×C173C2×C22⋊C414yes—
200C22×C173C23×C411yescharacteristic
201C22×C173C22×D411yescharacteristic
202C22×C173C22×D411yescharacteristic
203C22×C173C2×#4111yescharacteristic
204C22×C173C2×#4111yescharacteristic
205C22×C173H(5,𝔽2)11yescharacteristic
206C22×C1732−1+411yescharacteristic
207C2×D173—22no—
208C2×D173—22no—
209C2×D173—22no—
210C2×D173—22no—
211C2×D173—22no—
212C2×D173—22no—
213C2×D173C2×C22⋊C414yes—
214C4×C173—22no—
215C4×C173—22no—
216C4×C173—41no—
217C4×C173—41no—
218C4×C173—41no—
219Dic173—22no—
220Dic173—22no—
221Dic173—22no—
222Dic173—41no—
223Dic173—41no—
224Dic173—41no—
225Dic173C2×C22⋊C414yes—
226C2×C173—24no—
227C2×C173—42no—
228C2×C173C22×C22⋊C411yescharacteristic
229C2×C173C4×#4111yescharacteristic
230C2×C173#52011yescharacteristic
231C2×C173#54611yescharacteristic
232C2×C173#54611yescharacteristic
233C2×C173#54711yescharacteristic
234C2×C173#54711yescharacteristic
235D173—24no—
236C173#233911yescharacteristic, complemented, Sylow
237#2339—1731nomaximal, Sylow
238C22×#41—1731no—
239C2×C4×D4—1731no—
240C2×C4×D4—1731no—
241#391—1731no—
242#391—1731no—
243#386—1731no—
244#386—1731no—
245#386—1731no—
246#386D17311yescharacteristic, complemented
247#381—1731no—
248#381—1731no—
249#380—1731no—
250C4×C22⋊C4—1731no—
251C2×#95—1731no—
252C2×#95—1731no—
253C2×#41—3464no—
254C2×#41—3464no—
255C2×#41—1738no—
256C22×Q8—1731no—
257C22×D4—1731no—
258C22×D4—1731no—
259C22×D4C2×D17311yescharacteristic
260C23×C4—1731no—
261C23×C4—1731no—
262C23×C4—1731no—
263C23×C4—1731no—
264C23×C4—1731no—
265C23×C4—1731no—
266C23×C4C2×D17311yescharacteristic
267C4×D4—3464no—
268C4×D4—3464no—
269C2×C4⋊C4—1731no—
270C2×C4⋊C4—1731no—
271C2×C22⋊C4—1731no—
272C2×C22⋊C4—1731no—
273C2×C22⋊C4—1731no—
274C2×C22⋊C4—1731no—
275C2×C22⋊C4—1731no—
276C2×C22⋊C4C2×D17311yescharacteristic
277C2×C22⋊C4C2×D17311yescharacteristic
278C2×C22⋊C4C2×D17311yescharacteristic
279C2×C42—1731no—
280C2×C42—1731no—
281#95—1731no—
282#95—1731no—
283#95—1731no—
284#95—1731no—
285#95—1731no—
286#95—1731no—
287#95—1731no—
288#95—1731no—
289#95—1731no—
290#95—1731no—
291#95C2×D17311yescharacteristic
292#95C2×D17311yescharacteristic
293C24—1731no—
294C24C22×D17311yescharacteristic
295C24C22×D17311yescharacteristic
296#41—34632no—
297C2×Q8—3462no—
298C2×Q8—3462no—
299C2×Q8—1734no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—3462no—
303C2×D4—3462no—
304C2×D4—1734no—
305C2×D4—1734no—
306C2×D4—1734no—
307C2×D4—1734no—
308C2×D4C4×D17314yescomplemented
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—3462no—
313C22×C4—3462no—
314C22×C4—3462no—
315C22×C4—3462no—
316C22×C4—3462no—
317C22×C4—3462no—
318C22×C4—3462no—
319C22×C4—3462no—
320C22×C4—3462no—
321C22×C4—3462no—
322C22×C4—3462no—
323C22×C4—3462no—
324C22×C4—1731no—
325C22×C4—1731no—
326C22×C4—1731no—
327C22×C4—1734no—
328C22×C4—1734no—
329C22×C4—1731no—
330C22×C4—1731no—
331C22×C4—1731no—
332C22×C4—1734no—
333C22×C4—1731no—
334C22×C4—6921no—
335C22×C4—6921no—
336C22×C4—6921no—
337C22×C4C22×D17311yescharacteristic
338C22×C4C22×D17311yescharacteristic
339C22×C4C22×D17311yescharacteristic
340C22×C4C22×D17311yescharacteristic
341C22×C4C22×D17311yescharacteristic
342C4⋊C4—3462no—
343C4⋊C4—3462no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C22⋊C4—3462no—
348C22⋊C4—3462no—
349C22⋊C4—3462no—
350C22⋊C4—3462no—
351C22⋊C4—6921no—
352C42—3462no—
353C42—3462no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—3461no—
365C23—3461no—
366C23—3461no—
367C23—3461no—
368C23—1732no—
369C23—1732no—
370C23—1732no—
371C23C2×C4×D17312yes—
372C23C2×C4×D17312yes—
373C23D4×D17312yes—
374C23D4×D17312yes—
375C23C23×D17311yescharacteristic
376Q8—3468no—
377D4—28no—
378D4—3468no—
379D4—17316no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—44no—
394C2×C4—3468no—
395C2×C4—3468no—
396C2×C4—3461no—
397C2×C4—3461no—
398C2×C4—3461no—
399C2×C4—3461no—
400C2×C4—3461no—
401C2×C4—3461no—
402C2×C4—3468no—
403C2×C4—3461no—
404C2×C4—3461no—
405C2×C4—3461no—
406C2×C4—3461no—
407C2×C4—3461no—
408C2×C4—3461no—
409C2×C4—3461no—
410C2×C4—3461no—
411C2×C4—3461no—
412C2×C4—1732no—
413C2×C4—1732no—
414C2×C4—1732no—
415C2×C4—1732no—
416C2×C4—1732no—
417C2×C4—1732no—
418C2×C4—6924no—
419C2×C4—6924no—
420C2×C4—6924no—
421C2×C4C2×C4×D17312yes—
422C2×C4#1458382512yescomplemented
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—3462no—
437C22—3462no—
438C22—3462no—
439C22—3462no—
440C22—3462no—
441C22—3462no—
442C22—1734no—
443C22C22⋊C4×D17314yes—
444C22C22×C4×D17311yescharacteristic
445C22C2×D4×D17311yescharacteristic
446C22C2×D4×D17311yescharacteristic
447C22#41×D17311yescharacteristic
448C22C2×#1458382511yescharacteristic
449C22#1914972411yescharacteristic
450C22#1914972611yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—3462no—
457C4—3462no—
458C4—3462no—
459C4—1734no—
460C4—6921no—
461C4—6921no—
462C4—6921no—
463C2—24no—
464C2—42no—
465C2—3464no—
466C2C2×C22⋊C4×D17311yescharacteristic
467C2C4×#1458382511yescharacteristic
468C2#2601197011yescharacteristic
469C2#2601212811yescharacteristic
470C2#2601213011yescharacteristic
471C2#2601213911yescharacteristic
472C2#2601214111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#386×C173C2
socle, cosocleC23×C173C24
prime radicalC22×#14583825C2
hypercenter#386D173
hypocenterC173#2339
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D173C2semidirect
3. C2×#19149639C2semidirect
4. #26011395C2semidirect
5. #26011395C2semidirect
6. #26011371C2semidirect
7. #26011367C2semidirect
8. #26011367C2semidirect
9. #26011353C2semidirect
10. #26011352C2semidirect
11. #26011349C2semidirect
12. C22⋊C4×Dic173C2semidirect
13. C2×#19149586C2semidirect
14. #95×D173C2semidirect
15. #386×C173C2semidirect
18. C2×#14583825C4semidirect
60. C2×Dic346C2×C4semidirect
64. C2×C22⋊D173C2×C4semidirect
74. C22×Dic173C2×C4semidirect
75. C22×Dic173C2×C4semidirect
91. C2×C4×D173C2×C4semidirect
105. C2×D4×C173C2×C4semidirect
126. C22⋊D173C22⋊C4semidirect
150. C2×Dic173#41semidirect
185. C2×C4×C173#41semidirect
236. C173#2339semidirect
246. #386D173semidirect
308. C2×D4C4×D173semidirect
422. C2×C4#14583825semidirect

Maximal subgroups

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

subgroupindex
C22×#145838252
C2×C4×C22⋊D1732
C2×#191496392
#260113952
#260113712
#260113672
#260113532
#260113522
#260113492
C22⋊C4×Dic1732
C2×#191495862
#95×D1732
#386×C1732
#2339173