Finite Group Databasebeta

← Q8×#135×C67 #30740608

Subgroups — Q8×#135×C67 #30740608

Order 17152.

1786 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
1Q8×#41×C67C212yesmaximal, complemented
2C67×#3383C213yesmaximal, complemented
3Q8×Q16×C67C211yesmaximal, characteristic, complemented
4C67×#3381C216yesmaximal, complemented
5C67×#3380C213yesmaximal, complemented
6Q8×SD16×C67C212yesmaximal, complemented
7Q8×D8×C67C211yesmaximal, characteristic, complemented
8C67×#3160C213yesmaximal, complemented
9C67×#3076C216yesmaximal, complemented
10C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
11C4×#135×C67C213yesmaximal
12C2×#135×C67C2211yescharacteristic
13Q82×C67C2212yes—
14#557×C67C2216yes—
15#554×C67—26no—
16#554×C67C2216yes—
17D4×Q8×C67—22no—
18D4×Q8×C67C2212yes—
19#533×C67—26no—
20#533×C67C2216yes—
21C4×#41×C67C2216yes—
22C2×C4×Q8×C67—22no—
23C2×C4×Q8×C67C2211yescharacteristic
24#500×C67C2213yes—
25#500×C67C2213yes—
26#499×C67C2216yes—
27#499×C67C2216yes—
28#498×C67C2216yes—
29#479×C67C22112yes—
30#478×C67C22112yes—
31#477×C67C2216yes—
32#476×C67C2216yes—
33#475×C67C2216yes—
34#474×C67C2216yes—
35C8×Q8×C67C2211yescharacteristic
36C8×Q8×C67C2211yescharacteristic
37C8×Q8×C67C2213yes—
38C8×Q8×C67C2213yes—
39C4×Q16×C67C2213yes—
40C4×Q16×C67C2213yes—
41C4×SD16×C67C2216yes—
42C4×SD16×C67C2216yes—
43C4×D8×C67C2213yes—
44C4×D8×C67C2213yes—
45#427×C67C2216yes—
46#427×C67C2213yes—
47C2×C4⋊C8×C67C2213yes—
48#416×C67C2216yes—
49C2×C4×C8×C67C2213yes—
50C2×#41×C67C2312yes—
51C22×Q8×C67—22no—
52C22×Q8×C67D411yescharacteristic
53#135×C67Q818yesdirect product factor
54C2×Q16×C67C2311yescharacteristic
55C2×SD16×C67C2312yes—
56C2×D8×C67C2311yescharacteristic
57C22×C8×C67C2313yes—
58C22×C8×C67C2311yescharacteristic
59C4⋊Q8×C67—26no—
60C4⋊Q8×C67—26no—
61C4⋊Q8×C67C2316yes—
62C4⋊Q8×C67C2316yes—
63#125×C67—212no—
64#125×C67C23112yes—
65C22⋊Q8×C67—26no—
66C22⋊Q8×C67—26no—
67C22⋊Q8×C67—26no—
68C22⋊Q8×C67—26no—
69C22⋊Q8×C67—212no—
70C4×Q8×C67—22no—
71C4×Q8×C67—26no—
72C4×Q8×C67—26no—
73C4×Q8×C67—26no—
74C4×Q8×C67D411yescharacteristic
75C4×Q8×C67D413yes—
76C4×Q8×C67C2316yes—
77C4×Q8×C67C2316yes—
78C4×Q8×C67C2311yescharacteristic
79C4×Q8×C67C2313yes—
80C4×D4×C67—26no—
81C4×D4×C67—26no—
82C4×D4×C67C2316yes—
83C4×D4×C67C2316yes—
84#117×C67—26no—
85#117×C67C2316yes—
86C2×C4⋊C4×C67—26no—
87C2×C4⋊C4×C67D413yes—
88C2×C42×C67—26no—
89C2×C42×C67C2313yes—
90#107×C67C2316yes—
91#107×C67C2316yes—
92#107×C67C2313yes—
93#107×C67C2313yes—
94#106×C67C23112yes—
95#106×C67C2316yes—
96C4⋊C8×C67C2313yes—
97C4⋊C8×C67C2313yes—
98C4⋊C8×C67C2316yes—
99#103×C67C2316yes—
100#103×C67C2316yes—
101#102×C67C2316yes—
102#102×C67C2316yes—
103C4×C8×C67C2313yes—
104C4×C8×C67C2313yes—
105C4×C8×C67C2313yes—
106C4×C8×C67C2313yes—
107#41×C67—24no—
108#41×C67C2×Q818yescomplemented
109C2×Q8×C67—24no—
110C2×Q8×C67—26no—
111C2×Q8×C67—26no—
112C2×Q8×C67—48no—
113C2×Q8×C67C2×D413yes—
114C2×Q8×C67C2×D411yescharacteristic
115C2×Q8×C67C2412yes—
116C2×D4×C67—22no—
117C2×D4×C67C2412yes—
118C22×C4×C67—22no—
119C22×C4×C67—26no—
120C22×C4×C67—26no—
121C22×C4×C67C2×D413yes—
122C22×C4×C67C2×D413yes—
123C22×C4×C67C2411yescharacteristic
124Q16×C67C2×Q814yescomplemented
125SD16×C67C2×Q818yescomplemented
126D8×C67C2×Q814yescomplemented
127C2×C8×C67—26no—
128C2×C8×C67C2×Q814yescomplemented
129C2×C8×C67C2413yes—
130C2×C8×C67C2413yes—
131C2×C8×C67C2411yescharacteristic
132C2×C8×C67C2411yescharacteristic
133C4⋊C4×C67—212no—
134C4⋊C4×C67—26no—
135C4⋊C4×C67—26no—
136C4⋊C4×C67—26no—
137C4⋊C4×C67—26no—
138C4⋊C4×C67—212no—
139C4⋊C4×C67C2×D413yes—
140C4⋊C4×C67C2×D416yes—
141C4⋊C4×C67C2×D413yes—
142C4⋊C4×C67C2416yes—
143C4⋊C4×C67C2416yes—
144C22⋊C4×C67—26no—
145C22⋊C4×C67—26no—
146C22⋊C4×C67—26no—
147C42×C67—26no—
148C42×C67—26no—
149C42×C67—26no—
150C42×C67C2×D413yes—
151C42×C67C2×D413yes—
152C42×C67C2413yes—
153C42×C67C2413yes—
154C23×C67—22no—
155C23×C67C22×D411yescharacteristic
156Q8×C67—26no—
157Q8×C67—22no—
158Q8×C67—412no—
159Q8×C67—86no—
160Q8×C67#13514yesdirect product factor
161Q8×C67C22×Q814yes—
162D4×C67—28no—
163D4×C67—22no—
164D4×C67C22×Q814yes—
165C2×C4×C67—22no—
166C2×C4×C67—28no—
167C2×C4×C67—26no—
168C2×C4×C67—26no—
169C2×C4×C67—26no—
170C2×C4×C67—43no—
171C2×C4×C67—412no—
172C2×C4×C67—86no—
173C2×C4×C67C22×D413yes—
174C2×C4×C67C22×D413yes—
175C2×C4×C67C22×D413yes—
176C2×C4×C67C22×D413yes—
177C2×C4×C67C22×D411yescharacteristic
178C2×C4×C67C22×Q812yes—
179C2×C4×C672−1+412yes—
180C2×C4×C67C2511yescharacteristic
181C8×C67—23no—
182C8×C67—23no—
183C8×C67C22×Q812yes—
184C8×C67C22×Q812yes—
185C22×C67—21no—
186C22×C67—21no—
187C22×C67—24no—
188C22×C67—42no—
189C22×C67—42no—
190C22×C67D4×Q812yes—
191C22×C67C23×D411yescharacteristic
192C4×C67—23no—
193C4×C67—23no—
194C4×C67—24no—
195C4×C67—23no—
196C4×C67—46no—
197C4×C67—46no—
198C4×C67D4×Q812yes—
199C4×C67C2×#13516yes—
200C4×C67C23×Q811yescharacteristic
201C4×C67C2×2−1+411yescharacteristic
202Q8×#135C6711yesmaximal, characteristic, direct product factor, Sylow
203C2×C67—22no—
204C2×C67—44no—
205C2×C67C2×D4×Q811yescharacteristic
206C2×C67C22×#13511yescharacteristic
207C2×C67#358411yescharacteristic
208Q8×#41C2×C6712yescomplemented
209#3383C2×C6713yescomplemented
210Q8×Q16C2×C6711yescharacteristic, complemented
211#3381C2×C6716yescomplemented
212#3380C2×C6713yescomplemented
213Q8×SD16C2×C6712yescomplemented
214Q8×D8C2×C6711yescharacteristic, complemented
215#3160C2×C6713yescomplemented
216#3076C2×C6716yescomplemented
217C2×C8×Q8C2×C6711yescharacteristic, complemented
218C4×#135C2×C6713yes—
219C67Q8×#13511yescharacteristic, direct product factor, Sylow
220C2×#135C22×C6711yescharacteristic
221Q82C22×C6712yes—
222#557C22×C6716yes—
223#554—26no—
224#554C22×C6716yes—
225D4×Q8—22no—
226D4×Q8C22×C6712yes—
227#533—26no—
228#533C22×C6716yes—
229C4×#41C22×C6716yes—
230C2×C4×Q8—22no—
231C2×C4×Q8C22×C6711yescharacteristic
232#500C22×C6713yes—
233#500C22×C6713yes—
234#499C22×C6716yes—
235#499C22×C6716yes—
236#498C22×C6716yes—
237#479C22×C67112yes—
238#478C22×C67112yes—
239#477C22×C6716yes—
240#476C22×C6716yes—
241#475C22×C6716yes—
242#474C22×C6716yes—
243C8×Q8C22×C6711yescharacteristic
244C8×Q8C22×C6711yescharacteristic
245C8×Q8C22×C6713yes—
246C8×Q8C22×C6713yes—
247C4×Q16C22×C6713yes—
248C4×Q16C22×C6713yes—
249C4×SD16C22×C6716yes—
250C4×SD16C22×C6716yes—
251C4×D8C22×C6713yes—
252C4×D8C22×C6713yes—
253#427C22×C6716yes—
254#427C22×C6713yes—
255C2×C4⋊C8C22×C6713yes—
256#416C22×C6716yes—
257C2×C4×C8C22×C6713yes—
258C2×#41C23×C6712yes—
259C22×Q8—22no—
260C22×Q8D4×C6711yescharacteristic
261#135Q8×C6718yesdirect product factor
262C2×Q16C23×C6711yescharacteristic
263C2×SD16C23×C6712yes—
264C2×D8C23×C6711yescharacteristic
265C22×C8C23×C6713yes—
266C22×C8C23×C6711yescharacteristic
267C4⋊Q8—26no—
268C4⋊Q8—26no—
269C4⋊Q8C23×C6716yes—
270C4⋊Q8C23×C6716yes—
271#125—212no—
272#125C23×C67112yes—
273C22⋊Q8—26no—
274C22⋊Q8—26no—
275C22⋊Q8—26no—
276C22⋊Q8—26no—
277C22⋊Q8—212no—
278C4×Q8—22no—
279C4×Q8—26no—
280C4×Q8—26no—
281C4×Q8—26no—
282C4×Q8D4×C6711yescharacteristic
283C4×Q8D4×C6713yes—
284C4×Q8C23×C6716yes—
285C4×Q8C23×C6716yes—
286C4×Q8C23×C6711yescharacteristic
287C4×Q8C23×C6713yes—
288C4×D4—26no—
289C4×D4—26no—
290C4×D4C23×C6716yes—
291C4×D4C23×C6716yes—
292#117—26no—
293#117C23×C6716yes—
294C2×C4⋊C4—26no—
295C2×C4⋊C4D4×C6713yes—
296C2×C42—26no—
297C2×C42C23×C6713yes—
298#107C23×C6713yes—
299#107C23×C6713yes—
300#107C23×C6716yes—
301#107C23×C6716yes—
302#106C23×C6716yes—
303#106C23×C67112yes—
304C4⋊C8C23×C6713yes—
305C4⋊C8C23×C6713yes—
306C4⋊C8C23×C6716yes—
307#103C23×C6716yes—
308#103C23×C6716yes—
309#102C23×C6716yes—
310#102C23×C6716yes—
311C4×C8C23×C6713yes—
312C4×C8C23×C6713yes—
313C4×C8C23×C6713yes—
314C4×C8C23×C6713yes—
315#41—24no—
316#41C2×Q8×C6718yescomplemented
317C2×Q8—24no—
318C2×Q8—26no—
319C2×Q8—26no—
320C2×Q8—48no—
321C2×Q8C2×D4×C6713yes—
322C2×Q8C2×D4×C6711yescharacteristic
323C2×Q8C24×C6712yes—
324C2×D4—22no—
325C2×D4C24×C6712yes—
326C22×C4—22no—
327C22×C4—26no—
328C22×C4—26no—
329C22×C4C2×D4×C6713yes—
330C22×C4C2×D4×C6713yes—
331C22×C4C24×C6711yescharacteristic
332Q16C2×Q8×C6714yescomplemented
333SD16C2×Q8×C6718yescomplemented
334D8C2×Q8×C6714yescomplemented
335C2×C8—26no—
336C2×C8C2×Q8×C6714yescomplemented
337C2×C8C24×C6713yes—
338C2×C8C24×C6713yes—
339C2×C8C24×C6711yescharacteristic
340C2×C8C24×C6711yescharacteristic
341C4⋊C4—212no—
342C4⋊C4—26no—
343C4⋊C4—26no—
344C4⋊C4—26no—
345C4⋊C4—26no—
346C4⋊C4—212no—
347C4⋊C4C2×D4×C6713yes—
348C4⋊C4C2×D4×C6716yes—
349C4⋊C4C2×D4×C6713yes—
350C4⋊C4C24×C6716yes—
351C4⋊C4C24×C6716yes—
352C22⋊C4—26no—
353C22⋊C4—26no—
354C22⋊C4—26no—
355C42—26no—
356C42—26no—
357C42—26no—
358C42C2×D4×C6713yes—
359C42C2×D4×C6713yes—
360C42C24×C6713yes—
361C42C24×C6713yes—
362C23—22no—
363C23C22×D4×C6711yescharacteristic
364Q8—26no—
365Q8—22no—
366Q8—412no—
367Q8—86no—
368Q8#135×C6714yesdirect product factor
369Q8C22×Q8×C6714yes—
370D4—28no—
371D4—22no—
372D4C22×Q8×C6714yes—
373C2×C4—22no—
374C2×C4—28no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—43no—
379C2×C4—412no—
380C2×C4—86no—
381C2×C4C22×D4×C6713yes—
382C2×C4C22×D4×C6713yes—
383C2×C4C22×D4×C6713yes—
384C2×C4C22×D4×C6711yescharacteristic
385C2×C4C22×D4×C6713yes—
386C2×C4C22×Q8×C6712yes—
387C2×C42−1+4×C6712yes—
388C2×C4C25×C6711yescharacteristic
389C8—23no—
390C8—23no—
391C8C22×Q8×C6712yes—
392C8C22×Q8×C6712yes—
393C22—21no—
394C22—21no—
395C22—24no—
396C22—42no—
397C22—42no—
398C22D4×Q8×C6712yes—
399C22C23×D4×C6711yescharacteristic
400C4—23no—
401C4—23no—
402C4—23no—
403C4—24no—
404C4—46no—
405C4—46no—
406C4D4×Q8×C6712yes—
407C4C2×#135×C6716yes—
408C4C23×Q8×C6711yescharacteristic
409C4C2×2−1+4×C6711yescharacteristic
410C2—22no—
411C2—44no—
412C2C2×D4×Q8×C6711yescharacteristic
413C2C22×#135×C6711yescharacteristic
414C2C67×#358411yescharacteristic
subgroupG/H
centerC2×C4×C67C22×D4
commutator, Frattini, cosocleC2×C4C25×C67
socleC22×C67C23×D4
prime radicalC2×#135×C67C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C2×C4×Q8×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. Q8×#41×C67C2semidirect
2. C67×#3383C2semidirect
3. Q8×Q16×C67C2semidirect
4. C67×#3381C2semidirect
5. C67×#3380C2semidirect
6. Q8×SD16×C67C2semidirect
7. Q8×D8×C67C2semidirect
8. C67×#3160C2semidirect
9. C67×#3076C2semidirect
10. C2×C8×Q8×C67C2semidirect
53. #135×C67Q8direct (N × H)
108. #41×C67C2×Q8semidirect
124. Q16×C67C2×Q8semidirect
125. SD16×C67C2×Q8semidirect
126. D8×C67C2×Q8semidirect
128. C2×C8×C67C2×Q8semidirect
160. Q8×C67#135direct (N × H)
202. Q8×#135C67direct (N × H)
208. Q8×#41C2×C67semidirect
209. #3383C2×C67semidirect
210. Q8×Q16C2×C67semidirect
211. #3381C2×C67semidirect
212. #3380C2×C67semidirect
213. Q8×SD16C2×C67semidirect
214. Q8×D8C2×C67semidirect
215. #3160C2×C67semidirect
216. #3076C2×C67semidirect
217. C2×C8×Q8C2×C67semidirect
219. C67Q8×#135direct (N × H)
261. #135Q8×C67direct (N × H)
316. #41C2×Q8×C67semidirect
332. Q16C2×Q8×C67semidirect
333. SD16C2×Q8×C67semidirect
334. D8C2×Q8×C67semidirect
336. C2×C8C2×Q8×C67semidirect
368. Q8#135×C67direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#41×C672
C67×#33832
Q8×Q16×C672
C67×#33812
C67×#33802
Q8×SD16×C672
Q8×D8×C672
C67×#31602
C67×#30762
C2×C8×Q8×C672
C4×#135×C672
Q8×#13567