Finite Group Databasebeta

← #46847949

Subgroups — #46847949

Order 34688.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic, complemented
2#31774566C211yesmaximal, characteristic, complemented
3#31774554C211yesmaximal, characteristic, complemented
4#31774285C211yesmaximal, characteristic, complemented
5#31774147C211yesmaximal, characteristic, complemented
6#31774126C211yesmaximal, characteristic, complemented
7#31774117C211yesmaximal, characteristic, complemented
8#31774095C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774042C211yesmaximal, characteristic, complemented
11#31774041C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13#31774004C211yesmaximal, characteristic, complemented
14#31773968C211yesmaximal, characteristic, complemented
15C2×#135×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18#22576192—24no—
19C2×#16864644C2211yescharacteristic
20C2×#16864643C2211yescharacteristic
21C2×#16864642C2211yescharacteristic
22C2×#16864641C2211yescharacteristic
23C2×#16864639C2211yescharacteristic
24C2×#16864637C2211yescharacteristic
25#22576156—21no—
26#22576156C2211yescharacteristic
27#22576147—21no—
28#22576147C2211yescharacteristic
29#22576139—21no—
30#22576139C2211yescharacteristic
31#22576134—21no—
32#22576134C2211yescharacteristic
33#22576131—21no—
34#22576131—21no—
35#22576131C2211yescharacteristic
36#22576131C2211yescharacteristic
37Q8×Dic271C2211yescharacteristic
38Q8×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271—21no—
41C4×C22⋊D271C2211yescharacteristic
42#22576092C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50C22⋊(C271⋊C8)C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52#22576070C2211yescharacteristic
53C2×#41×C271C2211yescharacteristic
54C2×#41×C271C2211yescharacteristic
55#135×C271—24no—
56C2×Q16×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic
58C2×SD16×C271C2211yescharacteristic
59C2×D8×C271C2211yescharacteristic
60C22×C8×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—41no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—41no—
67C2×D1084—21no—
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271D411yescharacteristic
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271C2311yescharacteristic
78#16864644—22no—
79#16864643—22no—
80#16864642—41no—
81#16864641—22no—
82#16864639—41no—
83#16864637—22no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C4⋊Dic271—21no—
87C4⋊Dic271—21no—
88C4⋊Dic271C2311yescharacteristic
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91Dic271⋊C4D411yescharacteristic
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95C22⋊Dic271—21no—
96C22⋊Dic271—21no—
97C22⋊Dic271D411yescharacteristic
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4D411yescharacteristic
100C4×Dic271—21no—
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103#41×C271—22no—
104#41×C271—22no—
105#41×C271—41no—
106#41×C271—41no—
107C2×Q8×C271C2311yescharacteristic
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271—21no—
110C2×D4×C271—21no—
111C2×D4×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271D414yescomplemented
117SD16×C271—22no—
118SD16×C271—22no—
119D8×C271—22no—
120C2×C8×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127Dic542—42no—
128D1084—21no—
129D1084—21no—
130D1084—42no—
131C22⋊D271—41no—
132C22⋊D271—42no—
133C22⋊D271—42no—
134C22⋊D271—81no—
135C22⋊D271—81no—
136C22⋊D271—81no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C2×Dic271C2×D411yescharacteristic
143C4×D271—21no—
144C4×D271—21no—
145C4×D271—42no—
146C4×D271—42no—
147C271⋊C8—21no—
148C271⋊C8—21no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271C2×D411yescharacteristic
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271C2×D412yes—
155Q8×C271C2×D412yes—
156D4×C271—21no—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—42no—
161D4×C271—42no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—42no—
167C2×C4×C271—42no—
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2411yescharacteristic
170C8×C271—21no—
171C8×C271C2×D412yes—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271C22×D411yescharacteristic
182C2×D271—21no—
183C2×D271—21no—
184C2×D271—21no—
185C2×D271—41no—
186C2×D271—41no—
187C2×D271—41no—
188C4×C271—21no—
189C4×C271—22no—
190C4×C271—22no—
191C4×C271C22×D411yescharacteristic
192C4×C271H(5,𝔽2)11yescharacteristic
193Dic271—21no—
194Dic271—21no—
195Dic271—21no—
196Dic271—41no—
197Dic271—41no—
198Dic271—41no—
199C2×C271—41no—
200C2×C271—81no—
201C2×C271—81no—
202C2×C271D4211yescharacteristic
203C2×C271#57511yescharacteristic
204C2×C271#57711yescharacteristic
205D271—41no—
206D271—81no—
207C271#327111yescharacteristic, complemented, Sylow
208#3271—2711nomaximal, Sylow
209C2×#137—2711no—
210C2×#135D27111yescharacteristic, complemented
211#550—2711no—
212#548—2711no—
213#496—2711no—
214#469—2711no—
215#465—2711no—
216#464—2711no—
217#460—2711no—
218#452—2711no—
219#449—2711no—
220#449—2711no—
221#448—2711no—
222#441—2711no—
223#435—2711no—
224C2×#41—2711no—
225C2×#41C2×D27111yescharacteristic
226C2×#41C2×D27111yescharacteristic
227C22×Q8—2711no—
228#137—5424no—
229#135—24no—
230C2×Q16—2711no—
231C2×Q16C2×D27111yescharacteristic
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16C2×D27111yescharacteristic
236C2×SD16C2×D27111yescharacteristic
237C2×D8—2711no—
238C2×D8C2×D27111yescharacteristic
239C2×M16—2711no—
240C22×C8C2×D27111yescharacteristic
241#127—2711no—
242#127—5421no—
243#124—2711no—
244#124—5421no—
245C22⋊Q8—2711no—
246C22⋊Q8—5421no—
247#121—2711no—
248#121—2711no—
249#121—2711no—
250#121—5421no—
251#121—5421no—
252#121—5421no—
253C4×Q8—2711no—
254C4×Q8—2711no—
255C4×D4—2711no—
256C4×D4—5421no—
257C4×D4—5421no—
258#106—2711no—
259C4⋊C8—2711no—
260#103—2711no—
261#103—2711no—
262#103—2711no—
263#102—2711no—
264#102—2711no—
265#102—2711no—
266C22⋊C8—2711no—
267C22⋊C8—2711no—
268#97—2711no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—5422no—
274#41—10841no—
275C2×Q8—2711no—
276C2×Q8—5421no—
277C2×Q8—5422no—
278C2×Q8—10841no—
279C2×Q8C22×D27111yescharacteristic
280C2×Q8C22×D27111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—2711no—
284C2×D4—2711no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4C22×D27111yescharacteristic
292C2×D4C22×D27111yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—2711no—
296C22×C4—5421no—
297C22×C4—5421no—
298C22×C4C22×D27111yescharacteristic
299Q16—5422no—
300Q16C22⋊D27114yescomplemented
301SD16—22no—
302SD16—22no—
303SD16—5422no—
304SD16—5422no—
305SD16—10841no—
306D8—22no—
307D8—10841no—
308M16—5422no—
309C2×C8—22no—
310C2×C8—2711no—
311C2×C8—2711no—
312C2×C8C22×D27111yescharacteristic
313C2×C8C22×D27111yescharacteristic
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—2711no—
317C4⋊C4—5421no—
318C4⋊C4—5421no—
319C4⋊C4—5421no—
320C22⋊C4—2711no—
321C22⋊C4—2711no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C42—2711no—
329C42—5421no—
330C42—5421no—
331C23—21no—
332C23—21no—
333C23—2711no—
334C23—5421no—
335C23D4×D27111yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—5421no—
339Q8—5421no—
340Q8—10842no—
341Q8C2×C22⋊D27112yes—
342Q8C2×C22⋊D27112yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—42no—
349D4—5421no—
350D4—5421no—
351D4—10842no—
352D4—10841no—
353D4—10842no—
354D4—10842no—
355D4—21681no—
356D4—21681no—
357D4—21681no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—2711no—
365C2×C4—2711no—
366C2×C4—2711no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—10842no—
373C2×C4—10842no—
374C2×C4D4×D27111yescharacteristic
375C2×C4C23×D27111yescharacteristic
376C8—21no—
377C8—5421no—
378C8—5421no—
379C8C2×C22⋊D27112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—5421no—
390C22—5421no—
391C22—5421no—
392C22—10841no—
393C22—10841no—
394C22—10841no—
395C22C2×D4×D27111yescharacteristic
396C4—21no—
397C4—22no—
398C4—22no—
399C4—5421no—
400C4—5421no—
401C4—5421no—
402C4—10841no—
403C4—10841no—
404C4—10841no—
405C4C22×C22⋊D27111yescharacteristic
406C4#2257620511yescharacteristic
407C2—41no—
408C2—81no—
409C2—81no—
410C2—10841no—
411C2—21681no—
412C2D4×C22⋊D27111yescharacteristic
413C2#3177469411yescharacteristic
414C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#135×C271C2
socleC22×C271C22×D4
hypercenterC2×#135D271
hypocenterC271#3271
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
2. #31774566C2semidirect
3. #31774554C2semidirect
4. #31774285C2semidirect
5. #31774147C2semidirect
6. #31774126C2semidirect
7. #31774117C2semidirect
8. #31774095C2semidirect
9. #31774059C2semidirect
10. #31774042C2semidirect
11. #31774041C2semidirect
12. #31774034C2semidirect
13. #31774004C2semidirect
14. #31773968C2semidirect
15. C2×#135×C271C2semidirect
116. Q16×C271D4semidirect
207. C271#3271semidirect
210. C2×#135D271semidirect
300. Q16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
#317745662
#317745542
#317742852
#317741472
#317741262
#317741172
#317740952
#317740592
#317740422
#317740412
#317740342
#317740042
#317739682
C2×#135×C2712
#3271271