Finite Group Databasebeta

← #10690078

Subgroups — #10690078

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#83296C211yesmaximal, characteristic
2#79040C211yesmaximal, characteristic, complemented
3C3×#1492C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic, complemented
5#69734C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67654C211yesmaximal, characteristic, complemented
8#8996—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#950—21no—
11C4×#235C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5233—21no—
15#5129—21no—
16#5122—21no—
17#5090C411yescharacteristic
18#5083C411yescharacteristic
19#5075C411yescharacteristic
20#5072C411yescharacteristic
21#5057—21no—
22#5054—21no—
23#4925C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4840—22no—
27#4835—21no—
28#4833—21no—
29C3×D4⋊C8—22no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32#4459—21no—
33#4456—21no—
34#3035—31no—
35#2925—31no—
36#1492D311yescharacteristic, complemented
37#1480—31no—
38#1476—31no—
39#1301—31no—
40#1277—31no—
41C2×C3×#41—21no—
42C2×#235D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C42C2311yescharacteristic
53#950—42no—
54#950—42no—
55C2×#216—21no—
56C2×#215—21no—
57C2×#214—21no—
58C2×#213—21no—
59C4×#67—21no—
60C4×#67—21no—
61#926D411yescharacteristic
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64C4×D12C2×C411yescharacteristic
65C4×D12D411yescharacteristic
66#872—21no—
67C42×D3—21no—
68C4×Dic6C2×C411yescharacteristic
69C4×Dic6D411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74#835—21no—
75#835—21no—
76#832—21no—
77#832—21no—
78#830—21no—
79(C2×D3)⋊C8—22no—
80(C2×D3)⋊C8—22no—
81#815—22no—
82C8×Dic3—22no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)D411yescharacteristic
89C4⋊(C3⋊C8)D411yescharacteristic
90C4×C3⋊C8—21no—
91C2×#135—61no—
92#532C2×D311yescharacteristic
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#423C2×D311yescharacteristic
105C2×C4⋊C8—31no—
106C2×C4⋊C8—31no—
107#416—61no—
108#408—61no—
109#405—61no—
110C2×C4×C8C2×D311yescharacteristic
111Q8⋊C8—61no—
112D4⋊C8—22no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—61no—
116C3×#41—41no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C2×C3×Q8—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—21no—
130#235—21no—
131#235—21no—
132#235—21no—
133#235—21no—
134C2×D12C22⋊C411yescharacteristic
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×Dic6C22⋊C411yescharacteristic
139C3×M16—41no—
140C3×M16—41no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42C22×C411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153#217—21no—
154#216—42no—
155#215—42no—
156#214—42no—
157#213—42no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—21no—
160C4⋊Dic3C22⋊C411yescharacteristic
161C4⋊Dic3C22⋊C411yescharacteristic
162Dic3⋊C4—21no—
163Dic3⋊C4—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×#41—21no—
177C2×#41—31no—
178#135—122no—
179#135—122no—
180C2×Q16—61no—
181C2×SD16—61no—
182C2×SD16—61no—
183C2×D8—61no—
184C2×M16—21no—
185C22×C8—21no—
186C22×C8—61no—
187C22×C8—61no—
188C4⋊Q8C4×D311yescharacteristic
189#127C4×D311yescharacteristic
190#124—21no—
191#121—21no—
192#121—21no—
193C4×Q8—31no—
194C4×Q8—31no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—31no—
198C4×D4—31no—
199C4×D4—61no—
200C4×D4—61no—
201#117—31no—
202#117—61no—
203C2×C42—61no—
204C2×C42C22×D311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—31no—
207C4⋊C8—31no—
208C4⋊C8—31no—
209C4⋊C8—31no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213C4⋊C8#6712yes—
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—41no—
227C23×C3#12311yescharacteristic
228C22×D3—21no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×D4—42no—
232C3×D4—41no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—81no—
237C3×D4—81no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#13511yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254#67—41no—
255#67—41no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259D12—21no—
260D12—21no—
261C4×D3—41no—
262C4×D3—41no—
263C4×D3—41no—
264C4×D3—41no—
265C4×D3—81no—
266Dic6—21no—
267Dic6—21no—
268C3×C8—42no—
269C3×C8—42no—
270C3⋊C8—41no—
271C3⋊C8—41no—
272C3⋊C8—41no—
273C3⋊C8—41no—
274C3⋊C8—81no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279#41—61no—
280#41—61no—
281#41—61no—
282#41—61no—
283C2×Q8—21no—
284C2×Q8—31no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—31no—
289C2×D4—41no—
290C2×D4—61no—
291C22×C4—41no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298Q16—122no—
299SD16—122no—
300SD16—122no—
301D8—122no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C22⋊C4—61no—
328C42—61no—
329C42—61no—
330C42—61no—
331C42C2×C4×D311yescharacteristic
332C42D3×D411yescharacteristic
333C42D3×D411yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3#38611yescharacteristic
341C22×C3#48211yescharacteristic
342C22×C3#48211yescharacteristic
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—81no—
350C3×C4#41611yescharacteristic
351C3×C4#41811yescharacteristic
352C3×C4#44311yescharacteristic
353C3×C4#44411yescharacteristic
354C3×C4#44911yescharacteristic
355C3×C4#44911yescharacteristic
356Dic3—41no—
357Dic3—41no—
358Dic3—41no—
359C23—41no—
360C23—61no—
361C23#88311yescharacteristic
362Q8—41no—
363Q8—41no—
364Q8—61no—
365Q8—61no—
366D4—42no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—61no—
372D4—61no—
373D4—81no—
374D4—81no—
375D4—121no—
376D4—121no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—241no—
391C2×C4D3×C22⋊C411yescharacteristic
392C2×C4#67⋊C411yescharacteristic
393C2×C4#88411yescharacteristic
394C2×C4#91111yescharacteristic
395C2×C4#91211yescharacteristic
396C2×C4#91411yescharacteristic
397C2×C4#91611yescharacteristic
398C2×C4C4×#6711yescharacteristic
399C2×C4#93711yescharacteristic
400C2×C4#93811yescharacteristic
401C8—42no—
402C8—42no—
403C8—121no—
404C8—121no—
405C8—121no—
406C8—121no—
407C8—241no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—161no—
411C2×C3#187511yescharacteristic
412C2×C3#187611yescharacteristic
413C2×C3#188111yescharacteristic
414D3—81no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22#477911yescharacteristic
424C22#478411yescharacteristic
425C22#495611yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—81no—
430C4—121no—
431C4—121no—
432C4—121no—
433C4#477011yescharacteristic
434C4#477211yescharacteristic
435C4#482311yescharacteristic
436C4#482411yescharacteristic
437C4#515811yescharacteristic
438C4#517211yescharacteristic
439C3#899611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—161no—
443C2—241no—
444C2#7001211yescharacteristic
445C2#7020111yescharacteristic
446C2#7039011yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1492C2
socleC22×C3#386
prime radical#83296C2
cosocleC2×C3×C42C23
hypercenter#1492D3
hypocenterC3#8996
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #79040C2semidirect
3. C3×#1492C2semidirect
4. #71800C2semidirect
5. #69734C2semidirect
6. #67714C2semidirect
7. #67654C2semidirect
36. #1492D3semidirect
439. C3#8996semidirect

Maximal subgroups

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

subgroupindex
#832962
#790402
C3×#14922
#718002
#697342
#677142
#676542
#89963