Finite Group Databasebeta

← #10695980

Subgroups — #10695980

Order 768.

566 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2562C211yesmaximal, characteristic, complemented
2#83019C211yesmaximal, characteristic, complemented
3#82310C211yesmaximal, characteristic
4#71735C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#69492C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10005—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#400—21no—
13C3×#394—21no—
14C3×#388—22no—
15C3×#380—21no—
16C3×#380—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5217—21no—
19#5067—21no—
20#5066—22no—
21#5064—21no—
22#5015—21no—
23#5014—22no—
24#5008—21no—
25#4968—21no—
26#4968—21no—
27#4967—21no—
28C2×#805—22no—
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31C4×#208C2211yescharacteristic
32#4543—22no—
33D4⋊(C3⋊C8)—22no—
34#3234—31no—
35#3109—31no—
36#2562D311yescharacteristic, complemented
37#1980—31no—
38#1528—31no—
39#1490—31no—
40#1285—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C42D412yes—
62C2×C3×C42C2311yescharacteristic
63C2×#213—22no—
64#930—21no—
65#930—21no—
66C2×#208—21no—
67#875—21no—
68#874D411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869D411yescharacteristic
72C3×#95—22no—
73C3×#95—22no—
74C3×#95—21no—
75#832—22no—
76#830—21no—
77#830—21no—
78#809—22no—
79#809—22no—
80#809—22no—
81#808—22no—
82#808—22no—
83#807—22no—
84#805—44no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)D412yes—
87#803—21no—
88#803—21no—
89#803D412yes—
90C4×C3⋊C8—21no—
91#532—31no—
92C2×#125C2×D311yescharacteristic
93C2×C4×D4C2×D311yescharacteristic
94#501—61no—
95#490—61no—
96#481—61no—
97#481—61no—
98#480—61no—
99#478—62no—
100#476—61no—
101#459—61no—
102#447—61no—
103#442—62no—
104#432—31no—
105#432—31no—
106C2×#104—62no—
107C4×M16—31no—
108#400—21no—
109#394—21no—
110#388—22no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4C2×D311yescharacteristic
114#330—62no—
115D4⋊C8—62no—
116C24×C3—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C2×#67—41no—
132#235—42no—
133#235—42no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—42no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—44no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—82no—
154C3×C42—42no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159#213—44no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164#208—22no—
165#208—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×#41—61no—
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172C2×D8—62no—
173C2×M16—61no—
174C4⋊Q8—31no—
175#127—31no—
176#125—21no—
177#125—21no—
178#125—21no—
179#125—21no—
180#124—61no—
181#121—61no—
182#121—61no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4#6711yescharacteristic
193C2×C4⋊C4#6711yescharacteristic
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C42#6712yes—
199C2×C42C22×D311yescharacteristic
200#107—62no—
201#106—62no—
202#106—62no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205#104—124no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#97—32no—
213#97—61no—
214#97—61no—
215C4×C8—61no—
216#95—22no—
217#95—22no—
218#95—21no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#12011yescharacteristic
227C22×D3—41no—
228C3×D4—22no—
229C3×D4—22no—
230C3×D4—41no—
231C3×D4—41no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4—82no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12711yescharacteristic
255#67—81no—
256#67—81no—
257C2×Dic3—41no—
258D12—42no—
259D12—42no—
260D12—42no—
261C4×D3—82no—
262Dic6—42no—
263C3⋊C8—42no—
264C3⋊C8—42no—
265C3⋊C8—42no—
266C24—21no—
267#41—122no—
268#41—122no—
269C2×Q8—61no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—42no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—121no—
286C22×C4C2×#6711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289D8—124no—
290M16—62no—
291M16—121no—
292C2×C8—62no—
293C2×C8—62no—
294C2×C8—121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C22⋊C4—44no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—82no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—42no—
315C42D3×D411yescharacteristic
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42#23711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3#39211yescharacteristic
329C22×C3#45311yescharacteristic
330C22×C3#45411yescharacteristic
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4—81no—
343C3×C4#45412yes—
344Dic3—81no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—121no—
353C23#95311yescharacteristic
354Q8—122no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—241no—
363D4—241no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—121no—
382C2×C4—242no—
383C2×C4#93711yescharacteristic
384C2×C4#93711yescharacteristic
385C2×C4#93811yescharacteristic
386C2×C4#93911yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4#94011yescharacteristic
389C8—122no—
390C8—122no—
391C8—122no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—82no—
395C2×C3#200311yescharacteristic
396C2×C3#200611yescharacteristic
397C2×C3#201011yescharacteristic
398D3—161no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22#522311yescharacteristic
412C22#524111yescharacteristic
413C22#524511yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—241no—
424C4#506212yes—
425C3#1000511yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—481no—
430C2#7171511yescharacteristic
431C2#7173011yescharacteristic
432C2#7187911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2562C2
socleC22×C3#392
prime radical#82310C2
hypercenter#2562D3
hypocenterC3#10005
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2562

Complements

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

NH ≅ G/Ntype
1. C3×#2562C2semidirect
2. #83019C2semidirect
4. #71735C2semidirect
5. #71599C2semidirect
6. #69492C2semidirect
7. #67621C2semidirect
36. #2562D3semidirect
425. C3#10005semidirect

Maximal subgroups

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

subgroupindex
C3×#25622
#830192
#823102
#717352
#715992
#694922
#676212
#100053