Finite Group Databasebeta

← #36356576

Subgroups — #36356576

Order 22144.

788 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149652C211yesmaximal, characteristic
2C2×C4×D692C211yesmaximal, characteristic, complemented
3C2×#19149639C211yesmaximal, characteristic, complemented
4C2×#19149639C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D173C211yesmaximal, characteristic, complemented
6#26011391C211yesmaximal, characteristic, complemented
7#26011388C211yesmaximal, characteristic, complemented
8#26011383C211yesmaximal, characteristic, complemented
9#26011369C211yesmaximal, characteristic, complemented
10#26011367C211yesmaximal, characteristic, complemented
11#26011366C211yesmaximal, characteristic, complemented
12#26011362C211yesmaximal, characteristic, complemented
13C4×#14583805C211yesmaximal, characteristic, complemented
14#26011338C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C22×C22⋊D173C2211yescharacteristic
19C23×Dic173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22C22×C4×D173C2211yescharacteristic
23C22×C4×D173C2211yescharacteristic
24#19149652C418yescomplemented
25C4×D692—24no—
26#19149639—24no—
27#19149639—24no—
28C2×C4⋊Dic173C2211yescharacteristic
29C2×C4⋊Dic173C2211yescharacteristic
30C2×#14583805C2211yescharacteristic
31C2×#14583805C2211yescharacteristic
32C2×#14583805C2211yescharacteristic
33C2×#14583805C2211yescharacteristic
34C4⋊C4×D173—24no—
35C2×C22⋊Dic173C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C2×#14583803C2211yescharacteristic
39C2×#14583803C2211yescharacteristic
40C2×#14583803C2211yescharacteristic
41C2×#14583803C2211yescharacteristic
42C2×C4×Dic173C2211yescharacteristic
43C2×C4×Dic173C2211yescharacteristic
44C2×C4×Dic173C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47#19149586C2211yescharacteristic
48#19149586C2211yescharacteristic
49C2×C4⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52C2×C22⋊C4×C173C2211yescharacteristic
53C2×C42×C173C2211yescharacteristic
54#95×C173C2211yescharacteristic
55#95×C173C2211yescharacteristic
56C23×D173C2311yescharacteristic
57C23×D173C2311yescharacteristic
58C2×D692—22no—
59C2×D692—22no—
60C2×D692C2×C414yescomplemented
61C2×C22⋊D173—22no—
62C2×C22⋊D173—22no—
63C2×C22⋊D173—22no—
64C2×C22⋊D173—22no—
65C2×C22⋊D173C2×C414yescomplemented
66C2×C22⋊D173C2×C414yescomplemented
67C22×Dic173—22no—
68C22×Dic173—22no—
69C22×Dic173—41no—
70C22×Dic173C2311yescharacteristic
71C22×Dic173C2311yescharacteristic
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C2×C4×D173—22no—
77C2×C4×D173—22no—
78C2×C4×D173—22no—
79C2×C4×D173—22no—
80C2×C4×D173—22no—
81C2×C4×D173—22no—
82C2×C4×D173—22no—
83C2×C4×D173—22no—
84C2×C4×D173—41no—
85C2×C4×D173—41no—
86C2×C4×D173—41no—
87C2×C4×D173C2×C414yescomplemented
88C4⋊Dic173—22no—
89C4⋊Dic173—22no—
90#14583805—22no—
91#14583805—22no—
92#14583805C2×C414yescomplemented
93#14583805D414yescomplemented
94C22⋊Dic173—41no—
95#14583803—22no—
96#14583803—22no—
97#14583803—22no—
98#14583803—41no—
99#14583803C2×C414yescomplemented
100#14583803D414yescomplemented
101C4×Dic173—22no—
102C4×Dic173—22no—
103C4×Dic173—22no—
104C24×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108C22×C4×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111C4⋊C4×C173—22no—
112C22⋊C4×C173—22no—
113C22⋊C4×C173—22no—
114C22⋊C4×C173C2×C414yescomplemented
115C42×C173—22no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—21no—
122C22×D173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173C22×C412yes—
127C22×D173C22×C412yes—
128C22×D173C2×D412yes—
129C22×D173#4112yes—
130D692—28no—
131C22⋊D173—28no—
132C22⋊D173—28no—
133C2×Dic173—21no—
134C2×Dic173—21no—
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—21no—
145C2×Dic173—44no—
146C2×Dic173C22×C412yes—
147C2×Dic173C22×C412yes—
148C2×Dic173C2×D412yes—
149C2×Dic173C2×D412yes—
150C2×Dic173C2×D412yes—
151C2×Dic173#4112yes—
152C4×D173—28no—
153C4×D173—44no—
154C4×D173—44no—
155C4×D173—44no—
156C23×C173—21no—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—21no—
160C23×C173—21no—
161C23×C173—21no—
162C23×C173C22×C412yes—
163C23×C173C2411yescharacteristic
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×C173—21no—
175C2×C4×C173—21no—
176C2×C4×C173—21no—
177C2×C4×C173—21no—
178C2×C4×C173C22×C412yes—
179C2×C4×C173C22×C412yes—
180C2×C4×C173C2×D412yes—
181C2×C4×C173C2×D412yes—
182C22×C173—22no—
183C22×C173—22no—
184C22×C173—22no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173C23×C411yescharacteristic
190C22×C173C22×D411yescharacteristic
191C22×C173C22×D411yescharacteristic
192C22×C173C2×#4111yescharacteristic
193C22×C173C2×#4111yescharacteristic
194C22×C173H(5,𝔽2)11yescharacteristic
195C22×C1732−1+411yescharacteristic
196C2×D173—22no—
197C2×D173—22no—
198C2×D173—22no—
199C2×D173—22no—
200C2×D173—22no—
201C2×D173—22no—
202C2×D173—22no—
203C2×D173—22no—
204C2×D173—22no—
205C2×D173—41no—
206C2×D173—41no—
207C2×D173—41no—
208C2×D173—41no—
209C2×D173C4×D414yescomplemented
210C4×C173—22no—
211C4×C173—22no—
212C4×C173—22no—
213C4×C173—22no—
214C4×C173—41no—
215C4×C173—41no—
216Dic173—22no—
217Dic173—22no—
218Dic173—22no—
219Dic173—41no—
220Dic173—41no—
221Dic173C4×D414yescomplemented
222C2×C173—42no—
223C2×C173C2×C4×D411yescharacteristic
224C2×C173C2×C4×D411yescharacteristic
225C2×C173#52011yescharacteristic
226C2×C173D4211yescharacteristic
227C2×C173#54711yescharacteristic
228C2×C173#54811yescharacteristic
229C2×C173#55311yescharacteristic
230D173—24no—
231D173—42no—
232C173#236111yescharacteristic, complemented, Sylow
233#2361—1731nomaximal, Sylow
234C2×#121—1731no—
235C2×C4×D4—1731no—
236C2×C4×D4—1731no—
237C2×C4×D4—1731no—
238C22×C4⋊C4—1731no—
239#390—1731no—
240#389—1731no—
241#388—1731no—
242#388D17311yescharacteristic, complemented
243#386—1731no—
244#386—1731no—
245#385—1731no—
246#385—1731no—
247C4×C4⋊C4—1731no—
248C4×C22⋊C4—1731no—
249C22×D4—1731no—
250C22×D4—1731no—
251C22×D4—1731no—
252C23×C4—1731no—
253C23×C4—1731no—
254C23×C4—1731no—
255C23×C4—1731no—
256C23×C4—1731no—
257#121—1738no—
258C4×D4—3464no—
259C4×D4—3464no—
260C4×D4—3464no—
261C2×C4⋊C4—3464no—
262C2×C4⋊C4—1731no—
263C2×C4⋊C4—1731no—
264C2×C4⋊C4—1731no—
265C2×C4⋊C4—1731no—
266C2×C4⋊C4—1731no—
267C2×C4⋊C4—1731no—
268C2×C4⋊C4C2×D17311yescharacteristic
269C2×C22⋊C4—1731no—
270C2×C22⋊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—
281C2×C42—1731no—
282C2×C42C2×D17311yescharacteristic
283#95—1731no—
284#95—1731no—
285#95—1731no—
286#95—1731no—
287#95C2×D17311yescharacteristic
288#95C2×D17311yescharacteristic
289C24—1731no—
290C24—1731no—
291C24C22×D17311yescharacteristic
292C2×D4—3462no—
293C2×D4—3462no—
294C2×D4—3462no—
295C2×D4—3462no—
296C2×D4—3462no—
297C2×D4—3462no—
298C2×D4—1734no—
299C2×D4—1734no—
300C2×D4—1734no—
301C22×C4—3462no—
302C22×C4—3462no—
303C22×C4—3462no—
304C22×C4—3462no—
305C22×C4—3462no—
306C22×C4—3462no—
307C22×C4—3462no—
308C22×C4—3462no—
309C22×C4—3462no—
310C22×C4—3462no—
311C22×C4—1731no—
312C22×C4—1731no—
313C22×C4—1731no—
314C22×C4—1731no—
315C22×C4—1731no—
316C22×C4—1734no—
317C22×C4—1731no—
318C22×C4—6921no—
319C22×C4—6921no—
320C22×C4—6921no—
321C22×C4—6921no—
322C22×C4C22×D17311yescharacteristic
323C22×C4C22×D17311yescharacteristic
324C22×C4C22×D17311yescharacteristic
325C22×C4C22×D17311yescharacteristic
326C22×C4C22×D17311yescharacteristic
327C22×C4C22×D17311yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—3462no—
330C4⋊C4—3462no—
331C4⋊C4—3462no—
332C4⋊C4—3462no—
333C4⋊C4—1734no—
334C4⋊C4—1734no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—3462no—
338C22⋊C4—3462no—
339C22⋊C4—3462no—
340C22⋊C4—1734no—
341C22⋊C4—1734no—
342C22⋊C4—6921no—
343C22⋊C4—6921no—
344C22⋊C4C4×D17314yescomplemented
345C42—22no—
346C42—3462no—
347C42—3462no—
348C42—3462no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—3461no—
356C23—3461no—
357C23—3461no—
358C23—3461no—
359C23—3461no—
360C23—3461no—
361C23—3461no—
362C23—3461no—
363C23—3461no—
364C23—3461no—
365C23—1732no—
366C23—1732no—
367C23—1732no—
368C23—1732no—
369C23C2×C4×D17312yes—
370C23C23×D17311yescharacteristic
371D4—3468no—
372D4—3468no—
373D4—3468no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
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—3468no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—3461no—
394C2×C4—3461no—
395C2×C4—3461no—
396C2×C4—3461no—
397C2×C4—3461no—
398C2×C4—3461no—
399C2×C4—3461no—
400C2×C4—3461no—
401C2×C4—1732no—
402C2×C4—1732no—
403C2×C4—1732no—
404C2×C4—1732no—
405C2×C4—1732no—
406C2×C4—1732no—
407C2×C4—6924no—
408C2×C4—6924no—
409C2×C4—6924no—
410C2×C4—6924no—
411C2×C4C2×C4×D17312yes—
412C2×C4C2×C4×D17312yes—
413C2×C4D4×D17312yes—
414C2×C4D4×D17312yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—3462no—
423C22—3462no—
424C22—3462no—
425C22—3462no—
426C22—3462no—
427C22—3462no—
428C22—3462no—
429C22—3462no—
430C22—3462no—
431C22—1734no—
432C22—6921no—
433C22—6921no—
434C22—6921no—
435C22—6921no—
436C22C22×C4×D17311yescharacteristic
437C22C2×D4×D17311yescharacteristic
438C22C2×D4×D17311yescharacteristic
439C22#41×D17311yescharacteristic
440C22#41×D17311yescharacteristic
441C22#1914972511yescharacteristic
442C22#1914972611yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—3462no—
450C4—3462no—
451C4—3462no—
452C4—1734no—
453C4—6921no—
454C4—6921no—
455C2—42no—
456C2—3464no—
457C2—6922no—
458C2C4×D4×D17311yescharacteristic
459C2C4×D4×D17311yescharacteristic
460C2#2601197111yescharacteristic
461C2#2601212511yescharacteristic
462C2#2601213911yescharacteristic
463C2#2601214911yescharacteristic
464C2#2601219211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149652C2
hypercenter#388D173
hypocenterC173#2361
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D692C2semidirect
3. C2×#19149639C2semidirect
4. C2×#19149639C2semidirect
5. C2×C4⋊C4×D173C2semidirect
6. #26011391C2semidirect
7. #26011388C2semidirect
8. #26011383C2semidirect
9. #26011369C2semidirect
10. #26011367C2semidirect
11. #26011366C2semidirect
12. #26011362C2semidirect
13. C4×#14583805C2semidirect
14. #26011338C2semidirect
15. #388×C173C2semidirect
24. #19149652C4semidirect
60. C2×D692C2×C4semidirect
65. C2×C22⋊D173C2×C4semidirect
66. C2×C22⋊D173C2×C4semidirect
87. C2×C4×D173C2×C4semidirect
92. #14583805C2×C4semidirect
93. #14583805D4semidirect
99. #14583803C2×C4semidirect
100. #14583803D4semidirect
114. C22⋊C4×C173C2×C4semidirect
209. C2×D173C4×D4semidirect
221. Dic173C4×D4semidirect
232. C173#2361semidirect
242. #388D173semidirect
344. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496522
C2×C4×D6922
C2×#191496392
C2×C4⋊C4×D1732
#260113912
#260113882
#260113832
#260113692
#260113672
#260113662
#260113622
C4×#145838052
#260113382
#388×C1732
#2361173