Finite Group Databasebeta

← #10695979

Subgroups — #10695979

Order 768.

566 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#83300C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic, complemented
3#73806C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic, complemented
5#69863C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10005—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C2×C3×#104—22no—
13C3×C4×M16C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5084—21no—
16#5081—21no—
17#5072—21no—
18#5071—21no—
19#5036—21no—
20#5034—21no—
21#5034—21no—
22#5033—21no—
23#5014—22no—
24#5009—22no—
25#4987—21no—
26#4987—21no—
27#4940—22no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4672—21no—
31#4659—21no—
32#4465—22no—
33#4459—22no—
34#3234—31no—
35#3109—31no—
36#2562—31no—
37#1980D311yescharacteristic, complemented
38#1528—31no—
39#1490—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C42D412yes—
54C2×C3×C42C2311yescharacteristic
55C2×#213—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—22no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#870—21no—
72C3×#104—44no—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×#95—21no—
76#832—22no—
77#832—22no—
78#832—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81#830—21no—
82#830—21no—
83#809—22no—
84#808—22no—
85#808—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yes—
89#803—21no—
90#803D412yes—
91#532C2×D311yescharacteristic
92C2×#125—31no—
93C2×C4×D4—31no—
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—22no—
107C4×M16C2×D311yescharacteristic
108#400—61no—
109#394—61no—
110#388—62no—
111#380—61no—
112#380—61no—
113C4×C4⋊C4C2×D311yescharacteristic
114#330—62no—
115D4⋊C8—62no—
116C23×D3—21no—
117C3×#41—42no—
118C3×#41—42no—
119C2×C3×Q8—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—41no—
124C22×C3×C4—22no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—42no—
135C2×C4×D3—42no—
136C2×C4×D3—42no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C3×M16—22no—
140C3×M16—41no—
141C2×C3×C8—41no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—42no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42—42no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#213—44no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—82no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×#41—21no—
170C22×D4—31no—
171C23×C4—61no—
172C2×D8—62no—
173C2×M16—21no—
174C4⋊Q8D1211yescharacteristic
175#127D1211yescharacteristic
176#125—61no—
177#125—61no—
178#125—61no—
179#125—61no—
180#124—21no—
181#121—21no—
182#121—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—62no—
198C2×C42#6712yes—
199C2×C42C22×D311yescharacteristic
200#107—62no—
201#106—62no—
202#106—62no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205#104—44no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#97—21no—
213#97—32no—
214#97—61no—
215C4×C8—21no—
216#95—21no—
217#95—62no—
218#95—62no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C22×D3—22no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C3×Q8—42no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—42no—
232C3×D4—81no—
233C3×D4—81no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—82no—
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12711yescharacteristic
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257D12—22no—
258D12—22no—
259D12—41no—
260D12—41no—
261C4×D3—82no—
262C4×D3—82no—
263C3×C8—42no—
264C3⋊C8—42no—
265C3⋊C8—42no—
266C24—61no—
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—122no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—122no—
283C22×C4—122no—
284C22×C4—121no—
285C22×C4—121no—
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4C2×#6711yescharacteristic
289D8—124no—
290M16—22no—
291M16—41no—
292C2×C8—41no—
293C2×C8—62no—
294C2×C8—62no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—42no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—242no—
314C42—42no—
315C42C2×D1211yescharacteristic
316C42#23511yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3#39211yescharacteristic
325C22×C3#45311yescharacteristic
326C22×C3#45411yescharacteristic
327C2×D3—42no—
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—81no—
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#45412yes—
342Dic3—81no—
343Dic3—81no—
344Dic3—81no—
345C23—41no—
346C23—62no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23#93711yescharacteristic
354Q8—42no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—62no—
359D4—62no—
360D4—81no—
361D4—81no—
362D4—121no—
363D4—121no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—242no—
382C2×C4—242no—
383C2×C4#88211yescharacteristic
384C2×C4#88211yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#93011yescharacteristic
388C2×C4#94011yescharacteristic
389C8—42no—
390C8—122no—
391C8—122no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—161no—
395C2×C3#200311yescharacteristic
396C2×C3#200611yescharacteristic
397C2×C3#201011yescharacteristic
398D3—82no—
399C22—21no—
400C22—21no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—122no—
405C22—122no—
406C22—122no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22#496011yescharacteristic
412C22#506211yescharacteristic
413C22#507711yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—82no—
420C4—81no—
421C4—241no—
422C4—241no—
423C4—241no—
424C4#482712yes—
425C3#1000511yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—242no—
430C2#6966411yescharacteristic
431C2#7038811yescharacteristic
432C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1980C2
socleC22×C3#392
prime radical#83300C2
hypercenter#1980D3
hypocenterC3#10005
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #83273C2semidirect
3. #73806C2semidirect
4. C3×#1980C2semidirect
5. #69863C2semidirect
6. #69855C2semidirect
7. #67621C2semidirect
37. #1980D3semidirect
425. C3#10005semidirect

Maximal subgroups

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

subgroupindex
#833002
#832732
#738062
C3×#19802
#698632
#698552
#676212
#100053