Finite Group Databasebeta

← #10690079

Subgroups — #10690079

Order 768.

484 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82525C211yesmaximal, characteristic
2#82515C211yesmaximal, characteristic, complemented
3C3×#1476C211yesmaximal, characteristic, complemented
4#70758C211yesmaximal, characteristic, complemented
5#69735C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67654C211yesmaximal, characteristic, complemented
8#8996—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#950—21no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5234—21no—
15#5031C411yescharacteristic
16#5028—21no—
17#5027C411yescharacteristic
18#5019C411yescharacteristic
19#5016—21no—
20#5015C411yescharacteristic
21#5002—21no—
22#5000—22no—
23#4997—21no—
24#4926—21no—
25#4925—21no—
26#4924C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C3×Q8⋊C8—21no—
30C3×D4⋊C8—21no—
31#4459—21no—
32#4459—21no—
33#4459—22no—
34#3035—31no—
35#2925—31no—
36#1492—31no—
37#1480—31no—
38#1476D311yescharacteristic, complemented
39#1301—31no—
40#1277—31no—
41C2×C3×#41D411yescharacteristic
42C2×#235—21no—
43C22×C3×C8—21no—
44C3×C4×Q8C2×C411yescharacteristic
45C3×C4×Q8D411yescharacteristic
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×#117—21no—
51C3×#117D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54#950—42no—
55#950—42no—
56C2×#216—21no—
57C2×#215—21no—
58C2×#214—21no—
59C2×#213—21no—
60#930—21no—
61#930—21no—
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65#875—21no—
66#874C2×C411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#869C2×C411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75#830—22no—
76#830—22no—
77#810—21no—
78#810—21no—
79#809—21no—
80#809—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)C2×C411yescharacteristic
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)D411yescharacteristic
87C4⋊(C3⋊C8)D411yescharacteristic
88C4⋊(C3⋊C8)D412yes—
89#803—22no—
90C4×C3⋊C8—22no—
91C2×#135—61no—
92#532—31no—
93C4×#41C2×D311yescharacteristic
94#464—61no—
95#463—61no—
96#462—31no—
97#461—31no—
98#460—31no—
99#459—31no—
100#433—62no—
101#432—61no—
102#431—61no—
103#431C2×D311yescharacteristic
104#423—31no—
105C2×C4⋊C8—31no—
106C2×C4⋊C8—31no—
107#416—61no—
108#408—61no—
109#405—61no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—21no—
112D4⋊C8—21no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—62no—
116C3×#41—21no—
117C3×#41—21no—
118C3×#41—21no—
119C3×#41—21no—
120C2×C3×Q8C22⋊C411yescharacteristic
121C2×C3×D4—21no—
122C2×C3×D4C22⋊C411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133#235—41no—
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—
142C2×C3×C8—41no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#216—42no—
158#215—42no—
159#214—42no—
160#213—42no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165#208—41no—
166#208—41no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×#41—61no—
177C2×#41D1211yescharacteristic
178#135—122no—
179#135—122no—
180C2×Q16—61no—
181C2×SD16—61no—
182C2×SD16—61no—
183C2×D8—61no—
184C2×M16—61no—
185C22×C8—21no—
186C22×C8—61no—
187C22×C8—61no—
188C4⋊Q8—31no—
189#127—31no—
190#124—61no—
191#121—61no—
192#121—61no—
193C4×Q8C4×D311yescharacteristic
194C4×Q8#6711yescharacteristic
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—61no—
198C4×D4—61no—
199C4×D4C4×D311yescharacteristic
200C4×D4#6711yescharacteristic
201#117—21no—
202#117D1211yescharacteristic
203C2×C42—21no—
204C2×C42C22×D311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—31no—
207C4⋊C8—31no—
208C4⋊C8—31no—
209C4⋊C8—31no—
210C4⋊C8—32no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213C4⋊C8—61no—
214#103—61no—
215#103—61no—
216#102—61no—
217#102—61no—
218C22⋊C8—21no—
219C22⋊C8—62no—
220C22⋊C8—62no—
221#97—62no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—62no—
226C23×C3—21no—
227C23×C3#12311yescharacteristic
228C22×D3—41no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
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#12111yescharacteristic
252C2×C3×C4#13511yescharacteristic
253C2×C3×C4#13511yescharacteristic
254C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
255C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
256#67—81no—
257#67—81no—
258C2×Dic3—41no—
259D12—42no—
260D12—41no—
261D12—41no—
262D12—41no—
263D12—41no—
264C4×D3—81no—
265C4×D3—81no—
266Dic6—41no—
267Dic6—41no—
268C3×C8—42no—
269C3×C8—81no—
270C3⋊C8—41no—
271C3⋊C8—41no—
272C3⋊C8—41no—
273C3⋊C8—41no—
274C3⋊C8—42no—
275#41—21no—
276#41—21no—
277#41—21no—
278#41—21no—
279#41—121no—
280#41—121no—
281#41—121no—
282#41—121no—
283C2×Q8—61no—
284C2×Q8(C2×D3)⋊C411yescharacteristic
285C2×D4—21no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—121no—
290C2×D4(C2×D3)⋊C411yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—121no—
295C22×C4C2×D1211yescharacteristic
296C22×C4#23711yescharacteristic
297C22×C4#23711yescharacteristic
298Q16—122no—
299SD16—122no—
300SD16—122no—
301D8—122no—
302M16—121no—
303M16—121no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—61no—
309C2×C8—62no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4(C2×D3)⋊C411yescharacteristic
322C4⋊C4(C2×D3)⋊C411yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42C2×C4×D311yescharacteristic
332C42D3×D411yescharacteristic
333C42D3×D411yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3#38611yescharacteristic
341C22×C3#48211yescharacteristic
342C22×C3#48211yescharacteristic
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4#41611yescharacteristic
353C3×C4#41811yescharacteristic
354C3×C4#44311yescharacteristic
355C3×C4#44411yescharacteristic
356C3×C4#44911yescharacteristic
357C3×C4#44911yescharacteristic
358Dic3—81no—
359C23—21no—
360C23—121no—
361C23#88611yescharacteristic
362Q8—21no—
363Q8—21no—
364Q8—121no—
365Q8—121no—
366D4—21no—
367D4—21no—
368D4—41no—
369D4—41no—
370D4—122no—
371D4—121no—
372D4—121no—
373D4—121no—
374D4—121no—
375D4—241no—
376D4—241no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4#67⋊C411yescharacteristic
392C2×C4#67⋊C411yescharacteristic
393C2×C4#88211yescharacteristic
394C2×C4#90411yescharacteristic
395C2×C4#90411yescharacteristic
396C2×C4#90811yescharacteristic
397C2×C4#90811yescharacteristic
398C2×C4C2×(C2×D3)⋊C411yescharacteristic
399C2×C4#93911yescharacteristic
400C2×C4#93911yescharacteristic
401C8—42no—
402C8—81no—
403C8—121no—
404C8—121no—
405C8—121no—
406C8—121no—
407C8—122no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—81no—
411C2×C3#187511yescharacteristic
412C2×C3#187611yescharacteristic
413C2×C3#188111yescharacteristic
414D3—161no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22#473711yescharacteristic
424C22#473711yescharacteristic
425C22#495811yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—241no—
433C4#473411yescharacteristic
434C4#473411yescharacteristic
435C4#482311yescharacteristic
436C4#482411yescharacteristic
437C4#511311yescharacteristic
438C4#513111yescharacteristic
439C3#899611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—81no—
443C2—481no—
444C2#6993311yescharacteristic
445C2#7029311yescharacteristic
446C2#7037711yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1476C2
socleC22×C3#386
prime radical#82525C2
cosocleC2×C3×C42C23
hypercenter#1476D3
hypocenterC3#8996
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #82515C2semidirect
3. C3×#1476C2semidirect
4. #70758C2semidirect
5. #69735C2semidirect
6. #67745C2semidirect
7. #67654C2semidirect
38. #1476D3semidirect
439. C3#8996semidirect

Maximal subgroups

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

subgroupindex
#825252
#825152
C3×#14762
#707582
#697352
#677452
#676542
#89963