Finite Group Databasebeta

← #10690080

Subgroups — #10690080

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#79021C211yesmaximal, characteristic
2#79000C211yesmaximal, characteristic, complemented
3#70426C211yesmaximal, characteristic, complemented
4#69735C211yesmaximal, characteristic, complemented
5#69734C211yesmaximal, characteristic, complemented
6C3×#1277C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#8996—31nomaximal, Sylow
9C2×#904—21no—
10#5477C2211yescharacteristic
11C4×#235C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5113—21no—
16#5110—21no—
17#5101—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4849C411yescharacteristic
21#4848C411yescharacteristic
22#4845—21no—
23#4844—21no—
24#4843C411yescharacteristic
25#4841C411yescharacteristic
26#4840—22no—
27#4835—21no—
28#4833—21no—
29#4483—21no—
30#4483—21no—
31#4459—21no—
32#4459—22no—
33#4456—21no—
34#3035—31no—
35#2925—31no—
36#1492—31no—
37#1480—31no—
38#1476—31no—
39#1301—31no—
40#1277D311yescharacteristic, complemented
41C2×#235—21no—
42C2×#235D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C4×#67—21no—
50C4×#67—21no—
51#926D411yescharacteristic
52C2×#208—21no—
53C2×Dic12—21no—
54#904—42no—
55#904—42no—
56C2×D24—21no—
57C2×#204—21no—
58C2×#204—21no—
59#875—21no—
60#874C2×C411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63C4×D12C2×C411yescharacteristic
64C4×D12D411yescharacteristic
65#872—21no—
66C42×D3—21no—
67#869C2×C411yescharacteristic
68C4×Dic6C2×C411yescharacteristic
69C4×Dic6D411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8C2×C411yescharacteristic
73C3×C4⋊C8C2×C411yescharacteristic
74C3×C4⋊C8D411yescharacteristic
75C3×C4⋊C8D411yescharacteristic
76C3×C4×C8—21no—
77C3×C4×C8—21no—
78#830—21no—
79#821—21no—
80#821—21no—
81(C2×D3)⋊C8—22no—
82(C2×D3)⋊C8—22no—
83#816—21no—
84#816—21no—
85#815—22no—
86C8×Dic3—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90C4×C3⋊C8—21no—
91C2×#135—61no—
92#532—31no—
93C4×#41—31no—
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—31no—
103#431—61no—
104#423—31no—
105C2×C4⋊C8C2×D311yescharacteristic
106C2×C4⋊C8C2×D311yescharacteristic
107#416—61no—
108#408—61no—
109#405—61no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—61no—
112D4⋊C8—62no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—61no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—21no—
120C2×#67—41no—
121#235—21no—
122#235—21no—
123#235—21no—
124#235—21no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12C22⋊C411yescharacteristic
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C2×Dic6C22⋊C411yescharacteristic
139C2×C3×C8—22no—
140C2×C3×C8—21no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C3×C42C22×C411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154#217—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163Dic3⋊C4—21no—
164Dic3⋊C4—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—21no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172Dic12—42no—
173D24—42no—
174#204—42no—
175#204—42no—
176C2×#41—31no—
177C2×#41—61no—
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—21no—
187C22×C8—21no—
188C4⋊Q8—31no—
189#127—31no—
190#124—61no—
191#121—61no—
192#121—61no—
193C4×Q8—31no—
194C4×Q8—31no—
195C4×D4—31no—
196C4×D4—31no—
197C4×D4—61no—
198C4×D4—61no—
199C4×D4—61no—
200C4×D4—61no—
201#117—31no—
202#117—61no—
203C2×C42—61no—
204C2×C42C22×D311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—32no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210C4⋊C8C4×D311yescharacteristic
211C4⋊C8C4×D311yescharacteristic
212C4⋊C8D1211yescharacteristic
213C4⋊C8D1211yescharacteristic
214#103—61no—
215#103—61no—
216#102—61no—
217#102—61no—
218C22⋊C8—62no—
219C22⋊C8—62no—
220C22⋊C8—61no—
221#97—62no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—62no—
225C4×C8—61no—
226C23×C3#12311yescharacteristic
227C22×D3—21no—
228C22×D3—41no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#13511yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242#67—41no—
243#67—41no—
244#67—81no—
245#67—81no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—41no—
250D12—21no—
251D12—21no—
252D12—42no—
253D12—41no—
254D12—41no—
255D12—41no—
256D12—41no—
257C4×D3—41no—
258C4×D3—41no—
259C4×D3—41no—
260C4×D3—41no—
261C4×D3—81no—
262C4×D3—81no—
263C4×D3—81no—
264Dic6—21no—
265Dic6—21no—
266Dic6—41no—
267Dic6—41no—
268C3×C8—42no—
269C3×C8—41no—
270C3×C8—41no—
271C3×C8—41no—
272C3×C8—41no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275#41—61no—
276#41—61no—
277#41—61no—
278#41—61no—
279#41—121no—
280#41—121no—
281#41—121no—
282#41—121no—
283C2×Q8—31no—
284C2×Q8—61no—
285C2×D4—31no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—121no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—121no—
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23911yescharacteristic
298Q16—122no—
299SD16—122no—
300SD16—122no—
301D8—122no—
302M16—121no—
303M16—121no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—62no—
316C2×C8—121no—
317C4⋊C4—31no—
318C4⋊C4—31no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C42—61no—
329C42—61no—
330C42—61no—
331C42C2×C4×D311yescharacteristic
332C42C2×D1211yescharacteristic
333C42D3×D411yescharacteristic
334C42D3×D411yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3#38611yescharacteristic
338C22×C3#48211yescharacteristic
339C22×C3#48211yescharacteristic
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4#41611yescharacteristic
350C3×C4#41811yescharacteristic
351C3×C4#44311yescharacteristic
352C3×C4#44411yescharacteristic
353C3×C4#44911yescharacteristic
354C3×C4#44911yescharacteristic
355Dic3—41no—
356Dic3—41no—
357Dic3—41no—
358Dic3—81no—
359C23—61no—
360C23—121no—
361C23#89411yescharacteristic
362Q8—61no—
363Q8—61no—
364Q8—121no—
365Q8—121no—
366D4—61no—
367D4—61no—
368D4—122no—
369D4—121no—
370D4—121no—
371D4—121no—
372D4—121no—
373D4—121no—
374D4—121no—
375D4—241no—
376D4—241no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4C4×D1211yescharacteristic
392C2×C4D3×C22⋊C411yescharacteristic
393C2×C4#88211yescharacteristic
394C2×C4#88411yescharacteristic
395C2×C4#89311yescharacteristic
396C2×C4#89511yescharacteristic
397C2×C4#91111yescharacteristic
398C2×C4#91411yescharacteristic
399C2×C4#91611yescharacteristic
400C2×C4#91911yescharacteristic
401C8—42no—
402C8—41no—
403C8—41no—
404C8—41no—
405C8—41no—
406C8—122no—
407C8—241no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3#187511yescharacteristic
411C2×C3#187611yescharacteristic
412C2×C3#188111yescharacteristic
413D3—81no—
414D3—161no—
415C22—21no—
416C22—21no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22#466511yescharacteristic
424C22#486611yescharacteristic
425C22#488511yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—121no—
430C4—121no—
431C4—121no—
432C4—241no—
433C4#470011yescharacteristic
434C4#471711yescharacteristic
435C4#477011yescharacteristic
436C4#478111yescharacteristic
437C4#480411yescharacteristic
438C4#481011yescharacteristic
439C3#899611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—241no—
443C2—481no—
444C2#6869511yescharacteristic
445C2#6888111yescharacteristic
446C2#6894711yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socleC22×C3#386
prime radical#79021C2
cosocleC2×C3×C42C23
hypercenter#1277D3
hypocenterC3#8996
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #79000C2semidirect
3. #70426C2semidirect
4. #69735C2semidirect
5. #69734C2semidirect
6. C3×#1277C2semidirect
7. #67707C2semidirect
40. #1277D3semidirect
439. C3#8996semidirect

Maximal subgroups

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

subgroupindex
#790212
#790002
#704262
#697352
#697342
C3×#12772
#677072
#89963