Finite Group Databasebeta

← #10691008

Subgroups — #10691008

Order 768.

508 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
1#82310C211yesmaximal, characteristic
2#82303C211yesmaximal, characteristic, complemented
3C3×#1762C211yesmaximal, characteristic, complemented
4#69798C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic, complemented
6#67662C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9163—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C3×#102—21no—
12C2×C3×#97C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5217—21no—
15#5207—21no—
16C2×#830—21no—
17#5015C411yescharacteristic, complemented
18#5015C411yescharacteristic, complemented
19#5014—21no—
20#5014—21no—
21#5012—21no—
22#5011—22no—
23D4×C3⋊C8—21no—
24#5008C411yescharacteristic, complemented
25#5008C411yescharacteristic, complemented
26#4968—21no—
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C3×#340—21no—
31#4464—22no—
32#4451—21no—
33#4451—21no—
34#3109—31no—
35#2974—31no—
36#1762D311yescharacteristic, complemented
37#1530—31no—
38#1502—31no—
39#1302—31no—
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C22×C3×C8—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4C2×C411yescharacteristic
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#213—21no—
57C2×#213—21no—
58#930—21no—
59#930—21no—
60C2×#208—22no—
61C22×C3⋊C8—22no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic
67C3×#102—24no—
68C3×#97—21no—
69C3×#97—21no—
70C3×#95—21no—
71#832—21no—
72#832—21no—
73#830—22no—
74#830—22no—
75#830—21no—
76#830—21no—
77#830D411yescharacteristic
78#830D411yescharacteristic
79#809—21no—
80#809—21no—
81#808—21no—
82#808—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87#803—21no—
88#803D412yes—
89C4×C3⋊C8—21no—
90#532—31no—
91C2×C4×D4C2×D311yescharacteristic
92#481—61no—
93#476—31no—
94#476—31no—
95#459—31no—
96#459—31no—
97#447—61no—
98#442—61no—
99#442—61no—
100#436—61no—
101#435—62no—
102C8×D4—61no—
103#432—31no—
104#432—31no—
105#431—31no—
106C2×#102—21no—
107#407—61no—
108C2×C22⋊C8—61no—
109C2×#97C2×D311yescharacteristic
110C4×C4⋊C4C2×D311yescharacteristic
111#340—21no—
112#329—62no—
113#327—61no—
114#327—61no—
115C24×C3—21no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C2×C3×D4C22⋊C411yescharacteristic
120C2×C3×D4C22⋊C411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—82no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—41no—
132#235—81no—
133#235—81no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C42—81no—
152C3×C42C22×C411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#213—42no—
157#213—42no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162#208—82no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—82no—
169C2×#41—61no—
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172C2×D8—61no—
173C2×D8—61no—
174C2×M16—62no—
175C22×C8—21no—
176C22×C8—62no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4C4×D311yescharacteristic
187C4×D4C4×D311yescharacteristic
188C4×D4D1211yescharacteristic
189C4×D4D1211yescharacteristic
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#106—61no—
196#106—61no—
197C4⋊C8—31no—
198C4⋊C8—31no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201#102—24no—
202#102—61no—
203#102—61no—
204#102—61no—
205#102—61no—
206C22⋊C8—31no—
207C22⋊C8—31no—
208C22⋊C8—62no—
209C22⋊C8—62no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#97—21no—
213#97—21no—
214#97—32no—
215#97—61no—
216C4×C8—61no—
217#95—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3#12111yescharacteristic
225C22×D3—41no—
226C3×D4—21no—
227C3×D4—21no—
228C3×D4—21no—
229C3×D4—42no—
230C3×D4#10412yes—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12311yescharacteristic
252#67—81no—
253#67—81no—
254C2×Dic3—41no—
255D12—41no—
256D12—41no—
257D12—81no—
258D12—81no—
259C4×D3—82no—
260Dic6—81no—
261C3×C8—42no—
262C3⋊C8—41no—
263C3⋊C8—42no—
264C3⋊C8—41no—
265C3⋊C8—41no—
266C3⋊C8—41no—
267C24—21no—
268#41—241no—
269#41—241no—
270C2×Q8—61no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—121no—
278C2×D4(C2×D3)⋊C411yescharacteristic
279C2×D4(C2×D3)⋊C411yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—121no—
288C22×C4D3×D411yescharacteristic
289C22×C4#23711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291D8—122no—
292D8—122no—
293M16—242no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—62no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—242no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4(C2×D3)⋊C411yescharacteristic
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C42—81no—
317C42C2×C4×D311yescharacteristic
318C42C2×D1211yescharacteristic
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3#38611yescharacteristic
332C22×C3#47111yescharacteristic
333C22×C3#47211yescharacteristic
334C2×D3—81no—
335C2×D3—81no—
336C2×D3—81no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—82no—
345C3×C4C2×#10411yescharacteristic
346C3×C4#42111yescharacteristic
347Dic3—81no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—81no—
354C23—121no—
355C23#93911yescharacteristic
356Q8—241no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—121no—
362D4—121no—
363D4—241no—
364D4—241no—
365D4—241no—
366D4—241no—
367D4#80512yes—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—121no—
384C2×C4—242no—
385C2×C4#67⋊C411yescharacteristic
386C2×C4#67⋊C411yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#88611yescharacteristic
389C2×C4C2×(C2×D3)⋊C411yescharacteristic
390C2×C4#93911yescharacteristic
391C8—42no—
392C8—121no—
393C8—122no—
394C8—121no—
395C8—121no—
396C8—121no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—42no—
400C2×C3—81no—
401C2×C3#187111yescharacteristic
402C2×C3#190411yescharacteristic
403C2×C3#190711yescharacteristic
404D3—161no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22#495811yescharacteristic
419C22#519911yescharacteristic
420C22#520311yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—82no—
429C4—241no—
430C4C2×#80511yescharacteristic
431C4#500511yescharacteristic
432C3#916311yescharacteristic, complemented, Sylow
433C2—21no—
434C2—21no—
435C2—42no—
436C2—81no—
437C2—481no—
438C2#6945311yescharacteristic
439C2#7030311yescharacteristic
440C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1762C2
socleC22×C3#386
prime radical#82310C2
cosocleC2×C3×C42C23
hypercenter#1762D3
hypocenterC3#9163
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #82303C2semidirect
3. C3×#1762C2semidirect
4. #69798C2semidirect
5. #69469C2semidirect
6. #67662C2semidirect
7. #67621C2semidirect
17. #5015C4semidirect
18. #5015C4semidirect
24. #5008C4semidirect
25. #5008C4semidirect
36. #1762D3semidirect
432. C3#9163semidirect

Maximal subgroups

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

subgroupindex
#823102
#823032
C3×#17622
#697982
#694692
#676622
#676212
#91633