Finite Group Databasebeta

← #46848024

Subgroups — #46848024

Order 34688.

482 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic, complemented
2#31774566C211yesmaximal, characteristic
3Q8×C22⋊D271C211yesmaximal, characteristic, complemented
4#31774268C211yesmaximal, characteristic, complemented
5#31774143C211yesmaximal, characteristic, complemented
6#31774140C211yesmaximal, characteristic, complemented
7#31774105C211yesmaximal, characteristic, complemented
8#31774095C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774058C211yesmaximal, characteristic, complemented
11#31774036C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13Q16×Dic271C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15C2×#137×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18#22576192—24no—
19C2×#16864644C2211yescharacteristic
20C2×#16864643C2211yescharacteristic
21C2×#16864642C2211yescharacteristic
22C2×#16864641C2211yescharacteristic
23C2×#16864641C2211yescharacteristic
24C2×#16864637C2211yescharacteristic
25#22576158—21no—
26#22576158C2211yescharacteristic
27#22576156—21no—
28#22576156C2211yescharacteristic
29#22576139—21no—
30#22576139C2211yescharacteristic
31#22576136—21no—
32#22576136C2211yescharacteristic
33#22576134—21no—
34#22576134C2211yescharacteristic
35#22576131—21no—
36#22576131C2211yescharacteristic
37Q8×Dic271C2211yescharacteristic
38Q8×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271—21no—
41C4×C22⋊D271C2211yescharacteristic
42#22576094C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576082C2211yescharacteristic
49#22576081C2211yescharacteristic
50C22⋊(C271⋊C8)C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52C8×Dic271C2211yescharacteristic
53C2×#41×C271C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic
55#137×C271—24no—
56C2×Q16×C271C2211yescharacteristic
57C2×Q16×C271C2211yescharacteristic
58C2×SD16×C271C2211yescharacteristic
59C2×SD16×C271C2211yescharacteristic
60C2×M16×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⋊D271D411yescharacteristic
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864644—22no—
78#16864643—22no—
79#16864642—41no—
80#16864641—22no—
81#16864641—41no—
82#16864637—22no—
83C2×C271⋊C8C2311yescharacteristic
84C2×C271⋊C8C2311yescharacteristic
85C4⋊Dic271—21no—
86C4⋊Dic271—21no—
87C4⋊Dic271C2311yescharacteristic
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93Dic271⋊C4D411yescharacteristic
94C22⋊Dic271—21no—
95C22⋊Dic271—21no—
96C22⋊Dic271D411yescharacteristic
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4D411yescharacteristic
100C4×Dic271—21no—
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103#41×C271—22no—
104#41×C271—41no—
105C2×Q8×C271—21no—
106C2×Q8×C271—22no—
107C2×Q8×C271—41no—
108C2×Q8×C271C2311yescharacteristic
109C2×Q8×C271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271D414yescomplemented
118SD16×C271—22no—
119SD16×C271—22no—
120M16×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—22no—
132C22⋊D271—42no—
133C22⋊D271—81no—
134C22⋊D271—81no—
135C22⋊D271—81no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—21no—
143C4×D271—21no—
144C4×D271—42no—
145C4×D271—42no—
146C271⋊C8—21no—
147C271⋊C8—21no—
148C23×C271—21no—
149C23×C271C2×D411yescharacteristic
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—21no—
154Q8×C271—42no—
155Q8×C271C2×D412yes—
156Q8×C271C2×D412yes—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271—42no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—42no—
166C2×C4×C271—42no—
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2411yescharacteristic
169C8×C271—21no—
170C8×C271C2×D412yes—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271C22×D411yescharacteristic
178C2×D271—21no—
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—41no—
184C4×C271—21no—
185C4×C271—22no—
186C4×C271—22no—
187C4×C271—41no—
188C4×C271C22×D411yescharacteristic
189C4×C271H(5,𝔽2)11yescharacteristic
190Dic271—21no—
191Dic271—21no—
192Dic271—41no—
193Dic271—41no—
194Dic271—41no—
195Dic271#13712yescomplemented
196C2×C271—41no—
197C2×C271—81no—
198C2×C271D4211yescharacteristic
199C2×C271C2×#13711yescharacteristic
200C2×C271#57711yescharacteristic
201D271—41no—
202D271—81no—
203C271#327811yescharacteristic, complemented, Sylow
204#3278—2711nomaximal, Sylow
205C2×#137—2711no—
206C2×#137D27111yescharacteristic, complemented
207#550—2711no—
208D4×Q8—2711no—
209#492—2711no—
210#468—2711no—
211#468—2711no—
212#462—2711no—
213#460—2711no—
214#452—2711no—
215#452—2711no—
216#448—2711no—
217#448—2711no—
218C4×Q16—2711no—
219#436—2711no—
220C2×#41—2711no—
221C2×#41C2×D27111yescharacteristic
222C22×Q8—2711no—
223C22×Q8C2×D27111yescharacteristic
224#137—24no—
225#137—5424no—
226C2×Q16—2711no—
227C2×Q16C2×D27111yescharacteristic
228C2×Q16C2×D27111yescharacteristic
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16C2×D27111yescharacteristic
234C2×SD16C2×D27111yescharacteristic
235C2×M16—2711no—
236C2×M16C2×D27111yescharacteristic
237C4⋊Q8—2711no—
238C4⋊Q8—5421no—
239#127—2711no—
240#127—5421no—
241C22⋊Q8—2711no—
242C22⋊Q8—2711no—
243C22⋊Q8—5421no—
244C22⋊Q8—5421no—
245#121—2711no—
246#121—2711no—
247#121—5421no—
248#121—5421no—
249C4×Q8—2711no—
250C4×Q8—2711no—
251C4×D4—2711no—
252C4×D4—5421no—
253C4×D4—5421no—
254#107—2711no—
255C4⋊C8—2711no—
256#103—2711no—
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#102—2711no—
261#102—2711no—
262C22⋊C8—2711no—
263C22⋊C8—2711no—
264C4×C8—2711no—
265#41—22no—
266#41—41no—
267#41—5422no—
268#41—10841no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2711no—
273C2×Q8—5421no—
274C2×Q8—5422no—
275C2×Q8—10841no—
276C2×Q8C22×D27111yescharacteristic
277C2×Q8C22×D27111yescharacteristic
278C2×Q8C22×D27111yescharacteristic
279C2×D4—21no—
280C2×D4—2711no—
281C2×D4—2711no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4—5421no—
286C2×D4—5421no—
287C2×D4C22×D27111yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—2711no—
291C22×C4—5421no—
292C22×C4—5421no—
293C22×C4C22×D27111yescharacteristic
294Q16—22no—
295Q16—5422no—
296Q16C22⋊D27114yescomplemented
297SD16—22no—
298SD16—22no—
299SD16—5422no—
300SD16—5422no—
301SD16—10841no—
302SD16—10841no—
303M16—22no—
304M16—5422no—
305C2×C8—2711no—
306C2×C8—2711no—
307C2×C8C22×D27111yescharacteristic
308C2×C8C22×D27111yescharacteristic
309C4⋊C4—2711no—
310C4⋊C4—2711no—
311C4⋊C4—2711no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C4⋊C4—5421no—
315C4⋊C4—5421no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C22⋊C4—2711no—
319C22⋊C4—2711no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C42—2711no—
325C42—5421no—
326C42—5421no—
327C23—21no—
328C23—2711no—
329C23—5421no—
330C23D4×D27111yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—42no—
336Q8—5421no—
337Q8—5421no—
338Q8—10842no—
339Q8C2×C22⋊D27112yes—
340Q8C2×C22⋊D27112yes—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—5422no—
345D4—5421no—
346D4—5421no—
347D4—10842no—
348D4—10842no—
349D4—21681no—
350D4—21681no—
351D4—21681no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—2711no—
360C2×C4—2711no—
361C2×C4—2711no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—5421no—
367C2×C4—10842no—
368C2×C4—10842no—
369C2×C4D4×D27111yescharacteristic
370C2×C4C23×D27111yescharacteristic
371C8—21no—
372C8—5421no—
373C8—5421no—
374C8C2×C22⋊D27112yes—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—5421no—
382C22—5421no—
383C22—5421no—
384C22—10841no—
385C22—10841no—
386C22—10841no—
387C22C2×D4×D27111yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—2712no—
393C4—5421no—
394C4—5421no—
395C4—10841no—
396C4—10841no—
397C4—10841no—
398C4C22×C22⋊D27111yescharacteristic
399C4#2257620511yescharacteristic
400C2—41no—
401C2—81no—
402C2—10841no—
403C2—21681no—
404C2D4×C22⋊D27111yescharacteristic
405C2#137×D27111yescharacteristic
406C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
prime radical#31774566C2
hypercenterC2×#137D271
hypocenterC271#3278
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
3. Q8×C22⋊D271C2semidirect
4. #31774268C2semidirect
5. #31774143C2semidirect
6. #31774140C2semidirect
7. #31774105C2semidirect
8. #31774095C2semidirect
9. #31774059C2semidirect
10. #31774058C2semidirect
11. #31774036C2semidirect
12. #31774034C2semidirect
13. Q16×Dic271C2semidirect
14. #31773973C2semidirect
15. C2×#137×C271C2semidirect
117. Q16×C271D4semidirect
195. Dic271#137semidirect
203. C271#3278semidirect
206. C2×#137D271semidirect
296. 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
Q8×C22⋊D2712
#317742682
#317741432
#317741402
#317741052
#317740952
#317740592
#317740582
#317740362
#317740342
Q16×Dic2712
#317739732
C2×#137×C2712
#3278271