Finite Group Databasebeta

← #10691734

Subgroups — #10691734

Order 768.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3205C211yesmaximal, characteristic
2#83277C211yesmaximal, characteristic, complemented
3#71806C211yesmaximal, characteristic, complemented
4#71766C211yesmaximal, characteristic, complemented
5C2×#4544C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#9273—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×#948—21no—
12C3×#493—21no—
13C3×#486—21no—
14C3×#459—22no—
15C3×#447C414yes—
16C3×#431C2211yescharacteristic
17#5234—21no—
18C2×#833—21no—
19C2×#833—21no—
20#5012—21no—
21#5011—22no—
22D4×C3⋊C8—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—21no—
28#4544—21no—
29#4544—21no—
30#4544—21no—
31#3205D311yescharacteristic
32#2935—31no—
33#1965—31no—
34#1547—31no—
35C2×#331—31no—
36#1301—31no—
37#1283—31no—
38C2×C3×#41—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4C2×C412yes—
42C2×C3×D8—24no—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127D411yescharacteristic
49C3×#124—21no—
50C3×#121—21no—
51C3×#121C2×C412yes—
52C3×C4×D4—21no—
53C3×C4×D4D412yes—
54C2×C3×C42C2311yescharacteristic
55#948—42no—
56#948—42no—
57C2×#208—22no—
58C2×#208—21no—
59C2×#208D411yescharacteristic
60C2×#208D411yescharacteristic
61C22×C3⋊C8—22no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×#102—24no—
66C3×C22⋊C8C2×C412yes—
67C3×C4×C8—21no—
68#833—42no—
69#833—42no—
70#830—22no—
71#830—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D411yescharacteristic
78C4⋊(C3⋊C8)D411yescharacteristic
79#803—21no—
80C4×C3⋊C8—21no—
81C2×#131—61no—
82#532C2×D311yescharacteristic
83C2×#127C2×D311yescharacteristic
84#493—21no—
85#486—21no—
86#459—22no—
87#447Dic314yes—
88#436—61no—
89#435—62no—
90C8×D4—61no—
91#431C2×D311yescharacteristic
92#423—31no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8—31no—
95C2×#100—61no—
96C2×#100—61no—
97#408—61no—
98C4×M16—31no—
99#332—61no—
100#331—61no—
101#331—61no—
102#331—61no—
103C24×C3—22no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C2×C3×Q8—21no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—81no—
123C2×C3×D4—81no—
124C2×C3×D4—82no—
125C2×C3×D4C22⋊C412yes—
126C22×C3×C4—22no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C3×D8—84no—
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4C22⋊C412yes—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—22no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#208—21no—
141#208—21no—
142#208—21no—
143#208—44no—
144#208—41no—
145#208—41no—
146#208D812yescomplemented
147C2×C3⋊C8—21no—
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—44no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—81no—
157C2×#41—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C2×Dic312yes—
161C2×D8—24no—
162#131—122no—
163#131—122no—
164C2×M16—31no—
165C2×M16—31no—
166C2×M16—62no—
167C2×M16—61no—
168C22×C8—62no—
169C22×C8—61no—
170C22×C8—61no—
171C4⋊Q8#6711yescharacteristic
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#127#6711yescharacteristic
177#124—21no—
178#121—21no—
179#121C2×Dic312yes—
180C4×D4—21no—
181C4×D4#6712yes—
182C2×C42C22×D311yescharacteristic
183C4⋊C8—21no—
184C4⋊C8—31no—
185C4⋊C8—31no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#102—24no—
192#100—122no—
193#100—122no—
194C22⋊C8—62no—
195C22⋊C8—62no—
196C22⋊C8C2×Dic312yes—
197#97—61no—
198C4×C8—21no—
199C4×C8—61no—
200C23×C3—22no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—42no—
204C23×C3—82no—
205C23×C3—82no—
206C23×C3C4×D411yescharacteristic
207C3×Q8—41no—
208C3×Q8—41no—
209C3×D4—42no—
210C3×D4—44no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—41no—
217C3×D4—82no—
218C3×D4—82no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4#13511yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C3×C8—42no—
242C3⋊C8—22no—
243C3⋊C8—41no—
244C3⋊C8—41no—
245C3⋊C8—41no—
246C3⋊C8—81no—
247C3⋊C8—81no—
248C24—22no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—82no—
270C2×D4#21712yes—
271C22×C4—22no—
272C22×C4D3×D411yescharacteristic
273C22×C4#23711yescharacteristic
274C22×C4C22×Dic311yescharacteristic
275D8—84no—
276M16—32no—
277M16—61no—
278M16—61no—
279M16—61no—
280M16—124no—
281M16—121no—
282M16—121no—
283C2×C8—22no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—124no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—241no—
294C4⋊C4—21no—
295C4⋊C4#21712yes—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C42D3×D411yescharacteristic
299C42#23711yescharacteristic
300C42C2×#6711yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—162no—
311C22×C3—162no—
312C22×C3#38611yescharacteristic
313C22×C3C4×D811yescharacteristic
314C22×C3#44211yescharacteristic
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4#44711yescharacteristic
320C3×C4#44911yescharacteristic
321C3×C4#45911yescharacteristic
322C3×C4#46311yescharacteristic
323C3×C4#47111yescharacteristic
324C3×C4#47211yescharacteristic
325C23—22no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—82no—
330C23—82no—
331C23D4×Dic311yescharacteristic
332Q8—41no—
333Q8—41no—
334D4—42no—
335D4—44no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—82no—
343D4—82no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4D3×D811yescharacteristic
357C2×C4#91111yescharacteristic
358C2×C4#91111yescharacteristic
359C2×C4#91211yescharacteristic
360C2×C4D4×Dic311yescharacteristic
361C2×C4#93511yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4C2×#21711yescharacteristic
366C8—42no—
367C8—62no—
368C8—121no—
369C8—121no—
370C8—121no—
371C8—241no—
372C8—241no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—82no—
376C2×C3—162no—
377C2×C3#187911yescharacteristic
378C2×C3#190711yescharacteristic
379C2×C3#191411yescharacteristic
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—162no—
389C22—162no—
390C22Dic3×D811yescharacteristic
391C22#515211yescharacteristic
392C22#522011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4#515011yescharacteristic
398C4#515311yescharacteristic
399C4#515611yescharacteristic
400C4#515811yescharacteristic
401C4#519911yescharacteristic
402C4#520011yescharacteristic
403C3#927311yescharacteristic, complemented, Sylow
404C2—21no—
405C2—21no—
406C2—82no—
407C2—162no—
408C2#7185511yescharacteristic
409C2#7203711yescharacteristic
410C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3205C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3205D3
hypocenterC3#9273
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
2. #83277C2semidirect
3. #71806C2semidirect
4. #71766C2semidirect
5. C2×#4544C2semidirect
6. #67745C2semidirect
7. #67744C2semidirect
146. #208D8semidirect
403. C3#9273semidirect

Maximal subgroups

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

subgroupindex
C3×#32052
#832772
#718062
#717662
C2×#45442
#677452
#677442
#92733