Finite Group Databasebeta

← #10698673

Subgroups — #10698673

Order 768.

460 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 91 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#82992C211yesmaximal, characteristic
2C2×C3×#329C211yesmaximal, characteristic
3#71726C211yesmaximal, characteristic
4#69827C211yesmaximal, characteristic, complemented
5#69816C211yesmaximal, characteristic, complemented
6C2×#4465C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10427—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13#5230—21no—
14C2×#835—21no—
15#5222—21no—
16C2×#832—21no—
17C2×#818—21no—
18C2×#817—21no—
19#4953—21no—
20#4949—21no—
21#4948—21no—
22#4948—21no—
23#4936C2211yescharacteristic
24#4935C2211yescharacteristic
25C2×#803C2211yescharacteristic
26C3×#329—22no—
27#4551—21no—
28#4551—21no—
29#4465—22no—
30#2820—31no—
31#1992—31no—
32#1959—31no—
33#1847—31no—
34C2×#330—31no—
35C2×#329D311yescharacteristic
36#1295—31no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×#124—21no—
42C3×#124—21no—
43C3×#124—21no—
44C3×#124—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C22×C3⋊C8—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×#97—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72#835—42no—
73#832—42no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#818—42no—
79#817—42no—
80#803—21no—
81#803—21no—
82C2×#125—31no—
83C2×#124C2×D311yescharacteristic
84C2×#107—61no—
85C2×#106—61no—
86C2×#103—61no—
87C2×#102—61no—
88C2×#97—31no—
89C2×#97C2×D311yescharacteristic
90C2×#97C2×D311yescharacteristic
91#399—61no—
92#397—61no—
93#396—61no—
94#396—61no—
95#395—61no—
96#394—61no—
97#384—31no—
98#382—31no—
99#340—61no—
100#340—61no—
101#330—62no—
102#329—22no—
103C24×C3—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C22×C3×C4—21no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—41no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135#217—81no—
136#217—81no—
137#217—81no—
138#217—81no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—42no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—81no—
155Dic3⋊C4—81no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8D1211yescharacteristic
160C22×D4D1211yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—61no—
164#125—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#124—21no—
169#124—21no—
170#124—21no—
171#124—21no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C42C22×D311yescharacteristic
190#107—122no—
191#106—122no—
192#103—122no—
193#102—122no—
194#97—21no—
195#97—21no—
196#97—21no—
197#97—61no—
198#97—61no—
199#97#6712yes—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3#12011yescharacteristic
212C3×Q8—42no—
213C3×Q8—81no—
214C3×D4—42no—
215C3×D4—81no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4#12311yescharacteristic
230C2×C3×C4#12411yescharacteristic
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C3×C8—24no—
247C3×C8—42no—
248C3⋊C8—42no—
249C24—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—21no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—121no—
264C22×C4C2×D1211yescharacteristic
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—62no—
274C2×C8—121no—
275C2×C8—121no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—122no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—241no—
292C4⋊C4—241no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—241no—
298C22⋊C4—241no—
299C22⋊C4—241no—
300C22⋊C4—241no—
301C42#23511yescharacteristic
302C42#23711yescharacteristic
303C42#23711yescharacteristic
304C42C2×#6711yescharacteristic
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3#39411yescharacteristic
313C22×C3#45311yescharacteristic
314C22×C3#45311yescharacteristic
315C22×C3#45411yescharacteristic
316C22×C3#45511yescharacteristic
317C22×C3#45611yescharacteristic
318C22×C3#45611yescharacteristic
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—81no—
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329Dic3—81no—
330Dic3—161no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23#88211yescharacteristic
339Q8—42no—
340Q8—81no—
341D4—42no—
342D4—81no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4#88311yescharacteristic
368C2×C4#88511yescharacteristic
369C2×C4#88611yescharacteristic
370C2×C4#93011yescharacteristic
371C2×C4#93811yescharacteristic
372C2×C4#93911yescharacteristic
373C8—24no—
374C8—42no—
375C8—122no—
376C2×C3—161no—
377C2×C3#165611yescharacteristic
378C2×C3#166011yescharacteristic
379C2×C3#166111yescharacteristic
380C2×C3#166411yescharacteristic
381C2×C3#203711yescharacteristic
382C2×C3#203811yescharacteristic
383C2×C3#203911yescharacteristic
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#474611yescharacteristic
392C22#474711yescharacteristic
393C22#482211yescharacteristic
394C22#482611yescharacteristic
395C22#482611yescharacteristic
396C22#482711yescharacteristic
397C22#496111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—241no—
406C4—241no—
407C4—241no—
408C4—241no—
409C4—481no—
410C3#1042711yescharacteristic, complemented, Sylow
411C2—161no—
412C2#6835111yescharacteristic
413C2#6835511yescharacteristic
414C2#6835711yescharacteristic
415C2#6835911yescharacteristic
416C2#7006611yescharacteristic
417C2#7036011yescharacteristic
418C2#7036111yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#329C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenterC2×#329D3
hypocenterC3#10427
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
4. #69827C2semidirect
5. #69816C2semidirect
6. C2×#4465C2semidirect
7. #67659C2semidirect
410. C3#10427semidirect

Maximal subgroups

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

subgroupindex
#829922
C2×C3×#3292
#717262
#698272
#698162
C2×#44652
#676592
#104273