Finite Group Databasebeta

← C67×#8992 #30688243

Subgroups — C67×#8992 #30688243

Order 17152.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3103C211yesmaximal, characteristic
2Q8×M16×C67C211yesmaximal, characteristic, complemented
3C67×#1980C211yesmaximal, characteristic
4C4×#103×C67C211yesmaximal, characteristic, complemented
5C67×#1492C211yesmaximal, characteristic
6C67×#1303C211yesmaximal, characteristic, complemented
7C67×#1278C211yesmaximal, characteristic, complemented
8#532×C67C2211yescharacteristic
9C2×C4×Q8×C67C2211yescharacteristic
10#481×C67—21no—
11#475×C67C411yescharacteristic, complemented
12#475×C67C411yescharacteristic, complemented
13#460×C67C411yescharacteristic, complemented
14#460×C67C411yescharacteristic, complemented
15#448×C67—21no—
16#446×C67—21no—
17#446×C67—22no—
18C8×Q8×C67—21no—
19C4×SD16×C67—21no—
20C4×SD16×C67—21no—
21#431×C67C2211yescharacteristic
22#423×C67—21no—
23#423×C67C2211yescharacteristic
24C2×#104×C67—21no—
25C2×#104×C67—21no—
26C2×#103×C67—21no—
27C4×M16×C67—21no—
28C4×M16×C67C2211yescharacteristic
29C2×C4×C8×C67C2211yescharacteristic
30C4×C4⋊C4×C67C2211yescharacteristic
31#340×C67—21no—
32D4⋊C8×C67—22no—
33C2×#41×C67—21no—
34C22×Q8×C67D411yescharacteristic
35C2×SD16×C67—21no—
36C2×SD16×C67—21no—
37C2×M16×C67—21no—
38C2×M16×C67D411yescharacteristic
39C2×M16×C67D411yescharacteristic
40C22×C8×C67—21no—
41C4⋊Q8×C67C2×C411yescharacteristic
42#127×C67C2×C411yescharacteristic
43#124×C67—21no—
44#121×C67—21no—
45#121×C67—21no—
46C4×Q8×C67—21no—
47C4×Q8×C67—21no—
48C4×Q8×C67C2×C411yescharacteristic
49C4×Q8×C67C2×C411yescharacteristic
50C4×Q8×C67D411yescharacteristic
51C4×Q8×C67D411yescharacteristic
52C4×D4×C67—21no—
53C4×D4×C67—21no—
54C2×C4⋊C4×C67—21no—
55C2×C4⋊C4×C67—21no—
56C2×C4⋊C4×C67D411yescharacteristic
57C2×C42×C67—21no—
58C2×C42×C67—21no—
59C2×C42×C67C2311yescharacteristic
60#106×C67—21no—
61#106×C67—21no—
62C4⋊C8×C67—22no—
63C4⋊C8×C67—22no—
64C4⋊C8×C67—21no—
65C4⋊C8×C67C2×C411yescharacteristic
66C4⋊C8×C67C2×C411yescharacteristic
67C4⋊C8×C67D412yes—
68#104×C67—42no—
69#104×C67—42no—
70#103×C67—24no—
71#103×C67—21no—
72#103×C67—21no—
73#102×C67—21no—
74#102×C67—21no—
75C22⋊C8×C67—21no—
76#97×C67—22no—
77#97×C67—21no—
78C4×C8×C67—22no—
79C4×C8×C67—21no—
80C4×C8×C67—21no—
81C4×C8×C67—21no—
82C4×C8×C67—21no—
83#95×C67—21no—
84#41×C67—41no—
85#41×C67—41no—
86#41×C67—41no—
87#41×C67—41no—
88C2×Q8×C67—21no—
89C2×Q8×C67—22no—
90C2×Q8×C67—21no—
91C2×Q8×C67—41no—
92C2×Q8×C67C22⋊C411yescharacteristic
93C2×Q8×C67C22⋊C411yescharacteristic
94C2×D4×C67—21no—
95C2×D4×C67—21no—
96C2×D4×C67—21no—
97C2×D4×C67—41no—
98C22×C4×C67—21no—
99C22×C4×C67—21no—
100C22×C4×C67—41no—
101C22×C4×C67—41no—
102C22×C4×C67C2×D411yescharacteristic
103C22×C4×C67C2×D411yescharacteristic
104C22×C4×C67#4111yescharacteristic
105SD16×C67—42no—
106SD16×C67—42no—
107M16×C67—21no—
108M16×C67—21no—
109M16×C67—21no—
110M16×C67—41no—
111M16×C67—41no—
112M16×C67SD1612yescomplemented
113C2×C8×C67—21no—
114C2×C8×C67—22no—
115C2×C8×C67—21no—
116C2×C8×C67—21no—
117C2×C8×C67—21no—
118C2×C8×C67—22no—
119C2×C8×C67—41no—
120C2×C8×C67—41no—
121C4⋊C4×C67—21no—
122C4⋊C4×C67—21no—
123C4⋊C4×C67—21no—
124C4⋊C4×C67—21no—
125C4⋊C4×C67—21no—
126C4⋊C4×C67—22no—
127C4⋊C4×C67—21no—
128C4⋊C4×C67—21no—
129C4⋊C4×C67—41no—
130C4⋊C4×C67C22⋊C411yescharacteristic
131C4⋊C4×C67C22⋊C411yescharacteristic
132C22⋊C4×C67—41no—
133C22⋊C4×C67—41no—
134C42×C67—21no—
135C42×C67—21no—
136C42×C67—21no—
137C42×C67—21no—
138C42×C67—41no—
139C42×C67—41no—
140C42×C67C22×C411yescharacteristic
141C42×C67C2×D411yescharacteristic
142C42×C67C2×D411yescharacteristic
143C42×C67#4111yescharacteristic
144C23×C67—41no—
145C23×C67#12111yescharacteristic
146Q8×C67—21no—
147Q8×C67—21no—
148Q8×C67—21no—
149Q8×C67—42no—
150Q8×C67—41no—
151Q8×C67—41no—
152Q8×C67#10412yescomplemented
153D4×C67—42no—
154D4×C67—41no—
155D4×C67—41no—
156D4×C67—41no—
157D4×C67—41no—
158D4×C67—81no—
159D4×C67—81no—
160C2×C4×C67—21no—
161C2×C4×C67—21no—
162C2×C4×C67—21no—
163C2×C4×C67—21no—
164C2×C4×C67—21no—
165C2×C4×C67—21no—
166C2×C4×C67—41no—
167C2×C4×C67—42no—
168C2×C4×C67—42no—
169C2×C4×C67—41no—
170C2×C4×C67—81no—
171C2×C4×C67—81no—
172C2×C4×C67—81no—
173C2×C4×C67—81no—
174C2×C4×C67—81no—
175C2×C4×C67C2×C22⋊C411yescharacteristic
176C2×C4×C67C4×D411yescharacteristic
177C2×C4×C67C4×D411yescharacteristic
178C2×C4×C67#12011yescharacteristic
179C2×C4×C67#12111yescharacteristic
180C2×C4×C67#12311yescharacteristic
181C2×C4×C67C2×SD1611yescharacteristic
182C2×C4×C67#13511yescharacteristic
183C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C67#13711yescharacteristic
185C8×C67—22no—
186C8×C67—42no—
187C8×C67—41no—
188C8×C67—41no—
189C8×C67—41no—
190C8×C67—42no—
191C22×C67—21no—
192C22×C67—21no—
193C22×C67—81no—
194C22×C67—81no—
195C22×C67—81no—
196C22×C67#38611yescharacteristic
197C22×C67#46011yescharacteristic
198C22×C67#46411yescharacteristic
199C4×C67—21no—
200C4×C67—22no—
201C4×C67—21no—
202C4×C67—21no—
203C4×C67—41no—
204C4×C67—41no—
205C4×C67—41no—
206C4×C67—82no—
207C4×C67—81no—
208C4×C67C2×#10411yescharacteristic
209C4×C67#42111yescharacteristic
210C4×C67C4×SD1611yescharacteristic
211C4×C67#44011yescharacteristic
212C4×C67#44811yescharacteristic
213C4×C67#44911yescharacteristic
214#8992C6711yesmaximal, characteristic, direct product factor, Sylow
215C2×C67—21no—
216C2×C67—21no—
217C2×C67—161no—
218C2×C67#187111yescharacteristic
219C2×C67#188011yescharacteristic
220C2×C67#188111yescharacteristic
221#3103C2×C6711yescharacteristic
222Q8×M16C2×C6711yescharacteristic, complemented
223#1980C2×C6711yescharacteristic
224C4×#103C2×C6711yescharacteristic, complemented
225#1492C2×C6711yescharacteristic
226#1303C2×C6711yescharacteristic, complemented
227#1278C2×C6711yescharacteristic, complemented
228C67#899211yescharacteristic, direct product factor, Sylow
229#532C22×C6711yescharacteristic
230C2×C4×Q8C22×C6711yescharacteristic
231#481—21no—
232#475C4×C6711yescharacteristic, complemented
233#475C4×C6711yescharacteristic, complemented
234#460C4×C6711yescharacteristic, complemented
235#460C4×C6711yescharacteristic, complemented
236#448—21no—
237#446—21no—
238#446—22no—
239C8×Q8—21no—
240C4×SD16—21no—
241C4×SD16—21no—
242#431C22×C6711yescharacteristic
243#423—21no—
244#423C22×C6711yescharacteristic
245C2×#104—21no—
246C2×#104—21no—
247C2×#103—21no—
248C4×M16—21no—
249C4×M16C22×C6711yescharacteristic
250C2×C4×C8C22×C6711yescharacteristic
251C4×C4⋊C4C22×C6711yescharacteristic
252#340—21no—
253D4⋊C8—22no—
254C2×#41—21no—
255C22×Q8D4×C6711yescharacteristic
256C2×SD16—21no—
257C2×SD16—21no—
258C2×M16—21no—
259C2×M16D4×C6711yescharacteristic
260C2×M16D4×C6711yescharacteristic
261C22×C8—21no—
262C4⋊Q8C2×C4×C6711yescharacteristic
263#127C2×C4×C6711yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C2×C4×C6711yescharacteristic
270C4×Q8C2×C4×C6711yescharacteristic
271C4×Q8D4×C6711yescharacteristic
272C4×Q8D4×C6711yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4D4×C6711yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C23×C6711yescharacteristic
281#106—21no—
282#106—21no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8C2×C4×C6711yescharacteristic
287C4⋊C8C2×C4×C6711yescharacteristic
288C4⋊C8D4×C6712yes—
289#104—42no—
290#104—42no—
291#103—24no—
292#103—21no—
293#103—21no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297#97—22no—
298#97—21no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304#95—21no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊C4×C6711yescharacteristic
314C2×Q8C22⋊C4×C6711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4C2×D4×C6711yescharacteristic
324C22×C4C2×D4×C6711yescharacteristic
325C22×C4#41×C6711yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16SD16×C6712yescomplemented
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—41no—
341C2×C8—41no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—41no—
351C4⋊C4C22⋊C4×C6711yescharacteristic
352C4⋊C4C22⋊C4×C6711yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—41no—
360C42—41no—
361C42C22×C4×C6711yescharacteristic
362C42C2×D4×C6711yescharacteristic
363C42C2×D4×C6711yescharacteristic
364C42#41×C6711yescharacteristic
365C23—41no—
366C23#121×C6711yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8#104×C6712yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4C2×C22⋊C4×C6711yescharacteristic
397C2×C4C4×D4×C6711yescharacteristic
398C2×C4C4×D4×C6711yescharacteristic
399C2×C4#120×C6711yescharacteristic
400C2×C4#121×C6711yescharacteristic
401C2×C4#123×C6711yescharacteristic
402C2×C4C2×SD16×C6711yescharacteristic
403C2×C4#135×C6711yescharacteristic
404C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
405C2×C4#137×C6711yescharacteristic
406C8—22no—
407C8—42no—
408C8—41no—
409C8—42no—
410C8—41no—
411C8—41no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#386×C6711yescharacteristic
418C22#460×C6711yescharacteristic
419C22#464×C6711yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—81no—
429C4C2×#104×C6711yescharacteristic
430C4#421×C6711yescharacteristic
431C4C4×SD16×C6711yescharacteristic
432C4#440×C6711yescharacteristic
433C4#448×C6711yescharacteristic
434C4#449×C6711yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2C67×#187111yescharacteristic
439C2C67×#188011yescharacteristic
440C2C67×#188111yescharacteristic
subgroupG/H
centerC2×C4×C67#120
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
socleC22×C67#386
prime radical#532×C67C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×M16×C67C2semidirect
4. C4×#103×C67C2semidirect
6. C67×#1303C2semidirect
7. C67×#1278C2semidirect
11. #475×C67C4semidirect
12. #475×C67C4semidirect
13. #460×C67C4semidirect
14. #460×C67C4semidirect
112. M16×C67SD16semidirect
152. Q8×C67#104semidirect
214. #8992C67direct (N × H)
222. Q8×M16C2×C67semidirect
224. C4×#103C2×C67semidirect
226. #1303C2×C67semidirect
227. #1278C2×C67semidirect
228. C67#8992direct (N × H)
232. #475C4×C67semidirect
233. #475C4×C67semidirect
234. #460C4×C67semidirect
235. #460C4×C67semidirect
333. M16SD16×C67semidirect
373. Q8#104×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31032
Q8×M16×C672
C67×#19802
C4×#103×C672
C67×#14922
C67×#13032
C67×#12782
#899267