Finite Group Databasebeta

← #10690083

Subgroups — #10690083

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
1C3×#3035C211yesmaximal, characteristic
2#82515C211yesmaximal, characteristic, complemented
3#71802C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic, complemented
5#71638C211yesmaximal, characteristic, complemented
6#67707C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8996—31nomaximal, Sylow
9C2×C3×#135—21no—
10C3×#532C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C3×#464—21no—
13C3×#463—21no—
14C3×#462C411yescharacteristic
15C3×#461C411yescharacteristic
16C3×#460C411yescharacteristic
17C3×#459C411yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#416—21no—
20#5234—21no—
21#5000—22no—
22#4997—21no—
23#4926—21no—
24#4925—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4×C3⋊C8C2211yescharacteristic
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32D4⋊(C3⋊C8)—21no—
33D4⋊(C3⋊C8)—22no—
34#3035D311yescharacteristic
35#2925—31no—
36#1492—31no—
37#1480—31no—
38#1476—31no—
39#1301—31no—
40#1277—31no—
41C2×C3×#41—21no—
42C2×C3×#41D411yescharacteristic
43C3×#135—42no—
44C3×#135—42no—
45C2×C3×Q16—21no—
46C2×C3×SD16—21no—
47C2×C3×SD16—21no—
48C2×C3×D8—21no—
49C22×C3×C8—21no—
50C3×C4⋊Q8C2×C411yescharacteristic
51C3×#127C2×C411yescharacteristic
52C3×#124—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×Q8C2×C411yescharacteristic
56C3×C4×Q8D411yescharacteristic
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4C2×C411yescharacteristic
62C3×C4×D4D411yescharacteristic
63C3×#117—21no—
64C3×#117D411yescharacteristic
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#208—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8C2×C411yescharacteristic
72C3×C4⋊C8C2×C411yescharacteristic
73C3×#103—21no—
74C3×#103—21no—
75C3×#102—21no—
76C3×#102—21no—
77#830—22no—
78#830—22no—
79#830—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D411yescharacteristic
84C4⋊(C3⋊C8)D411yescharacteristic
85C4⋊(C3⋊C8)D412yes—
86#803—22no—
87C4×C3⋊C8—22no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×#135—21no—
92#532C2×D311yescharacteristic
93C4×#41C2×D311yescharacteristic
94#464—21no—
95#463—21no—
96#462Dic311yescharacteristic
97#461Dic311yescharacteristic
98#460Dic311yescharacteristic
99#459Dic311yescharacteristic
100#433—62no—
101#432—61no—
102#431—31no—
103#431—61no—
104#423—31no—
105C2×C4⋊C8—31no—
106C2×C4⋊C8C2×D311yescharacteristic
107#416—21no—
108#408—61no—
109#405—61no—
110C2×C4×C8—31no—
111Q8⋊C8—61no—
112D4⋊C8—61no—
113D4⋊C8—61no—
114D4⋊C8—62no—
115D4⋊C8—61no—
116C3×#41—21no—
117C3×#41—21no—
118C3×#41—21no—
119C3×#41—21no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—21no—
125C2×C3×Q8C22⋊C411yescharacteristic
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—41no—
131C2×C3×D4C22⋊C411yescharacteristic
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—41no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C3×Q16—42no—
140C3×SD16—42no—
141C3×SD16—42no—
142C3×D8—42no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4C22⋊C411yescharacteristic
152C3×C4⋊C4C22⋊C411yescharacteristic
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42—21no—
159C3×C42—21no—
160C3×C42—21no—
161C3×C42C22×C411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165#208—41no—
166#208—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—22no—
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#6711yescharacteristic
178#135—42no—
179#135—42no—
180C2×Q16—21no—
181C2×SD16—21no—
182C2×SD16—21no—
183C2×D8—21no—
184C2×M16—61no—
185C22×C8—21no—
186C22×C8—61no—
187C22×C8—61no—
188C4⋊Q8C2×Dic311yescharacteristic
189#127C2×Dic311yescharacteristic
190#124—21no—
191#121—21no—
192#121—21no—
193C4×Q8C2×Dic311yescharacteristic
194C4×Q8#6711yescharacteristic
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4C2×Dic311yescharacteristic
200C4×D4#6711yescharacteristic
201#117—21no—
202#117#6711yescharacteristic
203C2×C42—21no—
204C2×C42C22×D311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—31no—
207C4⋊C8—31no—
208C4⋊C8—32no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212C4⋊C8C2×Dic311yescharacteristic
213C4⋊C8C2×Dic311yescharacteristic
214#103—21no—
215#103—21no—
216#102—21no—
217#102—21no—
218C22⋊C8—62no—
219C22⋊C8—62no—
220C22⋊C8—61no—
221#97—62no—
222C4×C8—62no—
223C4×C8—61no—
224C4×C8—61no—
225C4×C8—61no—
226C23×C3—21no—
227C23×C3—41no—
228C23×C3#12311yescharacteristic
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—42no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—81no—
243C3×D4—81no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4C2×C22⋊C411yescharacteristic
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#13511yescharacteristic
265C2×C3×C4#13511yescharacteristic
266C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C3×C8—41no—
269C3×C8—41no—
270C3⋊C8—42no—
271C3⋊C8—41no—
272C3⋊C8—41no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275#41—21no—
276#41—21no—
277#41—21no—
278#41—21no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8#21711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4#21711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4#23711yescharacteristic
296C22×C4#23711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298Q16—42no—
299SD16—42no—
300SD16—42no—
301D8—42no—
302M16—121no—
303M16—121no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—62no—
309C2×C8—61no—
310C2×C8—62no—
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—21no—
320C4⋊C4—21no—
321C4⋊C4#21711yescharacteristic
322C4⋊C4#21711yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42C22×Dic311yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3#38611yescharacteristic
344C22×C3#48211yescharacteristic
345C22×C3#48211yescharacteristic
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—81no—
353C3×C4#41611yescharacteristic
354C3×C4#41811yescharacteristic
355C3×C4#44311yescharacteristic
356C3×C4#44411yescharacteristic
357C3×C4#44911yescharacteristic
358C3×C4#44911yescharacteristic
359C23—21no—
360C23—41no—
361C23#93511yescharacteristic
362Q8—21no—
363Q8—21no—
364Q8—41no—
365Q8—41no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—41no—
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374D4—41no—
375D4—81no—
376D4—81no—
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—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4#93211yescharacteristic
392C2×C4#93211yescharacteristic
393C2×C4D4×Dic311yescharacteristic
394C2×C4D4×Dic311yescharacteristic
395C2×C4#93711yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#93911yescharacteristic
398C2×C4#95011yescharacteristic
399C2×C4#95011yescharacteristic
400C2×C4C2×#21711yescharacteristic
401C8—41no—
402C8—41no—
403C8—122no—
404C8—121no—
405C8—121no—
406C8—122no—
407C8—241no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—81no—
411C2×C3—161no—
412C2×C3#187511yescharacteristic
413C2×C3#187611yescharacteristic
414C2×C3#188111yescharacteristic
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#503511yescharacteristic
424C22#503511yescharacteristic
425C22#522011yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—81no—
433C4#503711yescharacteristic
434C4#503711yescharacteristic
435C4#519611yescharacteristic
436C4#519711yescharacteristic
437C4#521511yescharacteristic
438C4#523311yescharacteristic
439C3#899611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—81no—
443C2—161no—
444C2#7167811yescharacteristic
445C2#7208711yescharacteristic
446C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3035C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3035D3
hypocenterC3#8996
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
2. #82515C2semidirect
3. #71802C2semidirect
4. #71800C2semidirect
5. #71638C2semidirect
6. #67707C2semidirect
7. #67703C2semidirect
439. C3#8996semidirect

Maximal subgroups

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

subgroupindex
C3×#30352
#825152
#718022
#718002
#716382
#677072
#677032
#89963